Properties

Label 8-48e4-1.1-c14e4-0-0
Degree $8$
Conductor $5308416$
Sign $1$
Analytic cond. $1.26839\times 10^{7}$
Root an. cond. $7.72514$
Motivic weight $14$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.01e5·5-s − 3.18e6·9-s − 1.13e8·13-s − 2.22e8·17-s − 1.11e9·25-s − 1.75e10·29-s + 2.12e11·37-s − 1.07e12·41-s − 3.22e11·45-s + 4.13e11·49-s + 1.27e12·53-s − 1.06e13·61-s − 1.14e13·65-s − 4.09e13·73-s + 7.62e12·81-s − 2.24e13·85-s − 3.00e14·89-s − 3.40e14·97-s + 2.22e14·101-s + 5.78e14·109-s + 2.86e14·113-s + 3.62e14·117-s + 7.56e14·121-s − 1.98e14·125-s + 127-s + 131-s + 137-s + ⋯
L(s)  = 1  + 1.29·5-s − 2/3·9-s − 1.81·13-s − 0.541·17-s − 0.182·25-s − 1.01·29-s + 2.23·37-s − 5.50·41-s − 0.862·45-s + 0.610·49-s + 1.08·53-s − 3.40·61-s − 2.34·65-s − 3.70·73-s + 1/3·81-s − 0.700·85-s − 6.80·89-s − 4.21·97-s + 2.07·101-s + 3.16·109-s + 1.21·113-s + 1.20·117-s + 1.99·121-s − 0.415·125-s − 1.31·145-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(15-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s+7)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(5308416\)    =    \(2^{16} \cdot 3^{4}\)
Sign: $1$
Analytic conductor: \(1.26839\times 10^{7}\)
Root analytic conductor: \(7.72514\)
Motivic weight: \(14\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 5308416,\ (\ :7, 7, 7, 7),\ 1)\)

Particular Values

\(L(\frac{15}{2})\) \(\approx\) \(0.0008311143276\)
\(L(\frac12)\) \(\approx\) \(0.0008311143276\)
\(L(8)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad2 \( 1 \)
3$C_2$ \( ( 1 + p^{13} T^{2} )^{2} \)
good5$D_{4}$ \( ( 1 - 50556 T + 175654406 p^{2} T^{2} - 50556 p^{14} T^{3} + p^{28} T^{4} )^{2} \)
7$D_4\times C_2$ \( 1 - 59130445180 p T^{2} - \)\(70\!\cdots\!02\)\( p^{2} T^{4} - 59130445180 p^{29} T^{6} + p^{56} T^{8} \)
11$D_4\times C_2$ \( 1 - 568098483500 p^{3} T^{2} + \)\(23\!\cdots\!82\)\( p^{4} T^{4} - 568098483500 p^{31} T^{6} + p^{56} T^{8} \)
13$D_{4}$ \( ( 1 + 4370956 p T + 43763300787846 p^{2} T^{2} + 4370956 p^{15} T^{3} + p^{28} T^{4} )^{2} \)
17$D_{4}$ \( ( 1 + 111071532 T + 181324517337231014 T^{2} + 111071532 p^{14} T^{3} + p^{28} T^{4} )^{2} \)
19$D_4\times C_2$ \( 1 - 2139849100285849348 T^{2} + \)\(24\!\cdots\!58\)\( T^{4} - 2139849100285849348 p^{28} T^{6} + p^{56} T^{8} \)
23$D_4\times C_2$ \( 1 - 32033174640508842436 T^{2} + \)\(47\!\cdots\!86\)\( T^{4} - 32033174640508842436 p^{28} T^{6} + p^{56} T^{8} \)
29$D_{4}$ \( ( 1 + 8766353460 T + \)\(24\!\cdots\!62\)\( T^{2} + 8766353460 p^{14} T^{3} + p^{28} T^{4} )^{2} \)
31$D_4\times C_2$ \( 1 - \)\(25\!\cdots\!40\)\( T^{2} + \)\(27\!\cdots\!82\)\( T^{4} - \)\(25\!\cdots\!40\)\( p^{28} T^{6} + p^{56} T^{8} \)
37$D_{4}$ \( ( 1 - 106253480212 T + \)\(13\!\cdots\!14\)\( T^{2} - 106253480212 p^{14} T^{3} + p^{28} T^{4} )^{2} \)
41$D_{4}$ \( ( 1 + 536535822348 T + \)\(14\!\cdots\!98\)\( T^{2} + 536535822348 p^{14} T^{3} + p^{28} T^{4} )^{2} \)
43$D_4\times C_2$ \( 1 + \)\(19\!\cdots\!28\)\( T^{2} + \)\(19\!\cdots\!98\)\( T^{4} + \)\(19\!\cdots\!28\)\( p^{28} T^{6} + p^{56} T^{8} \)
47$D_4\times C_2$ \( 1 - \)\(30\!\cdots\!20\)\( T^{2} + \)\(26\!\cdots\!22\)\( T^{4} - \)\(30\!\cdots\!20\)\( p^{28} T^{6} + p^{56} T^{8} \)
53$D_{4}$ \( ( 1 - 635425281324 T + \)\(23\!\cdots\!82\)\( T^{2} - 635425281324 p^{14} T^{3} + p^{28} T^{4} )^{2} \)
59$D_4\times C_2$ \( 1 - \)\(14\!\cdots\!20\)\( T^{2} + \)\(10\!\cdots\!42\)\( T^{4} - \)\(14\!\cdots\!20\)\( p^{28} T^{6} + p^{56} T^{8} \)
61$D_{4}$ \( ( 1 + 5345601524780 T + \)\(26\!\cdots\!82\)\( T^{2} + 5345601524780 p^{14} T^{3} + p^{28} T^{4} )^{2} \)
67$D_4\times C_2$ \( 1 - \)\(51\!\cdots\!52\)\( T^{2} + \)\(11\!\cdots\!58\)\( T^{4} - \)\(51\!\cdots\!52\)\( p^{28} T^{6} + p^{56} T^{8} \)
71$D_4\times C_2$ \( 1 - \)\(27\!\cdots\!60\)\( T^{2} + \)\(32\!\cdots\!22\)\( T^{4} - \)\(27\!\cdots\!60\)\( p^{28} T^{6} + p^{56} T^{8} \)
73$D_{4}$ \( ( 1 + 20466380295196 T + \)\(33\!\cdots\!22\)\( T^{2} + 20466380295196 p^{14} T^{3} + p^{28} T^{4} )^{2} \)
79$D_4\times C_2$ \( 1 + \)\(42\!\cdots\!92\)\( T^{2} + \)\(26\!\cdots\!38\)\( T^{4} + \)\(42\!\cdots\!92\)\( p^{28} T^{6} + p^{56} T^{8} \)
83$D_4\times C_2$ \( 1 - \)\(15\!\cdots\!92\)\( T^{2} + \)\(16\!\cdots\!98\)\( T^{4} - \)\(15\!\cdots\!92\)\( p^{28} T^{6} + p^{56} T^{8} \)
89$D_{4}$ \( ( 1 + 150471761373852 T + \)\(95\!\cdots\!58\)\( T^{2} + 150471761373852 p^{14} T^{3} + p^{28} T^{4} )^{2} \)
97$D_{4}$ \( ( 1 + 170171711121532 T + \)\(20\!\cdots\!94\)\( T^{2} + 170171711121532 p^{14} T^{3} + p^{28} T^{4} )^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.689396334894756954644308241341, −8.463920957172987498946987103655, −8.186451767532952468889960391802, −7.39840324478045382009267869081, −7.32377562246987312713324672432, −7.06956543219123460152493401579, −6.86733229620046245444944649339, −6.08430308686107611684779360868, −6.01575910469085210661707621176, −5.63326346208081548332116352918, −5.63148422814953745782112773409, −4.86048689613360926059861877659, −4.69696671520313447383663569271, −4.56559398763748955633138341675, −3.91522528588754013504502049689, −3.55796866087183078776199125311, −2.91383719927888837673867207741, −2.73218005321556278512951633680, −2.71715053784259667154516690773, −1.80997973177734323634196694965, −1.80247592747598919723910259287, −1.65107556910749102924235847652, −1.07414453511241253199140325581, −0.33848446828065633390739647024, −0.00539301057318999739235620201, 0.00539301057318999739235620201, 0.33848446828065633390739647024, 1.07414453511241253199140325581, 1.65107556910749102924235847652, 1.80247592747598919723910259287, 1.80997973177734323634196694965, 2.71715053784259667154516690773, 2.73218005321556278512951633680, 2.91383719927888837673867207741, 3.55796866087183078776199125311, 3.91522528588754013504502049689, 4.56559398763748955633138341675, 4.69696671520313447383663569271, 4.86048689613360926059861877659, 5.63148422814953745782112773409, 5.63326346208081548332116352918, 6.01575910469085210661707621176, 6.08430308686107611684779360868, 6.86733229620046245444944649339, 7.06956543219123460152493401579, 7.32377562246987312713324672432, 7.39840324478045382009267869081, 8.186451767532952468889960391802, 8.463920957172987498946987103655, 8.689396334894756954644308241341

Graph of the $Z$-function along the critical line