Properties

Label 4-1008e2-1.1-c5e2-0-2
Degree $4$
Conductor $1016064$
Sign $1$
Analytic cond. $26136.1$
Root an. cond. $12.7148$
Motivic weight $5$
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  + 6·5-s + 98·7-s − 90·11-s + 768·13-s − 1.92e3·17-s − 2.24e3·19-s + 6.35e3·23-s − 654·25-s − 1.05e4·29-s + 3.31e3·31-s + 588·35-s + 2.10e3·37-s − 1.26e3·41-s + 5.76e3·43-s + 1.56e4·47-s + 7.20e3·49-s − 1.65e4·53-s − 540·55-s − 1.31e4·59-s − 5.79e3·61-s + 4.60e3·65-s + 5.61e4·67-s + 1.10e4·71-s − 8.53e4·73-s − 8.82e3·77-s + 1.96e4·79-s − 4.44e4·83-s + ⋯
L(s)  = 1  + 0.107·5-s + 0.755·7-s − 0.224·11-s + 1.26·13-s − 1.61·17-s − 1.42·19-s + 2.50·23-s − 0.209·25-s − 2.33·29-s + 0.618·31-s + 0.0811·35-s + 0.252·37-s − 0.117·41-s + 0.475·43-s + 1.03·47-s + 3/7·49-s − 0.807·53-s − 0.0240·55-s − 0.491·59-s − 0.199·61-s + 0.135·65-s + 1.52·67-s + 0.259·71-s − 1.87·73-s − 0.169·77-s + 0.353·79-s − 0.707·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1016064 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1016064 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1016064\)    =    \(2^{8} \cdot 3^{4} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(26136.1\)
Root analytic conductor: \(12.7148\)
Motivic weight: \(5\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1016064,\ (\ :5/2, 5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(2.923014791\)
\(L(\frac12)\) \(\approx\) \(2.923014791\)
\(L(\frac{7}{2})\) 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 \( 1 \)
7$C_1$ \( ( 1 - p^{2} T )^{2} \)
good5$D_{4}$ \( 1 - 6 T + 138 p T^{2} - 6 p^{5} T^{3} + p^{10} T^{4} \)
11$D_{4}$ \( 1 + 90 T + 51246 T^{2} + 90 p^{5} T^{3} + p^{10} T^{4} \)
13$D_{4}$ \( 1 - 768 T + 689558 T^{2} - 768 p^{5} T^{3} + p^{10} T^{4} \)
17$D_{4}$ \( 1 + 1926 T + 3761514 T^{2} + 1926 p^{5} T^{3} + p^{10} T^{4} \)
19$D_{4}$ \( 1 + 2248 T + 3007830 T^{2} + 2248 p^{5} T^{3} + p^{10} T^{4} \)
23$D_{4}$ \( 1 - 6354 T + 22960446 T^{2} - 6354 p^{5} T^{3} + p^{10} T^{4} \)
29$D_{4}$ \( 1 + 10572 T + 60053694 T^{2} + 10572 p^{5} T^{3} + p^{10} T^{4} \)
31$D_{4}$ \( 1 - 3312 T + 31130942 T^{2} - 3312 p^{5} T^{3} + p^{10} T^{4} \)
37$D_{4}$ \( 1 - 2104 T + 14492118 T^{2} - 2104 p^{5} T^{3} + p^{10} T^{4} \)
41$D_{4}$ \( 1 + 1266 T + 226048450 T^{2} + 1266 p^{5} T^{3} + p^{10} T^{4} \)
43$D_{4}$ \( 1 - 5768 T - 18440058 T^{2} - 5768 p^{5} T^{3} + p^{10} T^{4} \)
47$D_{4}$ \( 1 - 15612 T + 373470814 T^{2} - 15612 p^{5} T^{3} + p^{10} T^{4} \)
53$D_{4}$ \( 1 + 16512 T + 599616358 T^{2} + 16512 p^{5} T^{3} + p^{10} T^{4} \)
59$D_{4}$ \( 1 + 13140 T + 1459090998 T^{2} + 13140 p^{5} T^{3} + p^{10} T^{4} \)
61$D_{4}$ \( 1 + 5796 T + 1309453982 T^{2} + 5796 p^{5} T^{3} + p^{10} T^{4} \)
67$D_{4}$ \( 1 - 56116 T + 2298831942 T^{2} - 56116 p^{5} T^{3} + p^{10} T^{4} \)
71$D_{4}$ \( 1 - 11022 T - 822078402 T^{2} - 11022 p^{5} T^{3} + p^{10} T^{4} \)
73$D_{4}$ \( 1 + 85384 T + 5800143006 T^{2} + 85384 p^{5} T^{3} + p^{10} T^{4} \)
79$D_{4}$ \( 1 - 19620 T + 6216467102 T^{2} - 19620 p^{5} T^{3} + p^{10} T^{4} \)
83$D_{4}$ \( 1 + 44424 T + 7801188630 T^{2} + 44424 p^{5} T^{3} + p^{10} T^{4} \)
89$D_{4}$ \( 1 + 211218 T + 21434789410 T^{2} + 211218 p^{5} T^{3} + p^{10} T^{4} \)
97$D_{4}$ \( 1 - 44864 T + 3170652414 T^{2} - 44864 p^{5} T^{3} + p^{10} T^{4} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.365106844464828524164077208610, −8.788304226857600252919847954097, −8.660242259122220289873101619703, −8.496691580693985168973122679887, −7.65559443042212896723051636877, −7.42586523851789983103533892509, −6.85115370563623636349413323482, −6.59041533299849448563661684845, −5.86717995133336860686690317105, −5.78491089347295753035898415199, −4.99667737141553815695978103341, −4.71510774637363447902838894519, −4.00134588714583596209304140489, −3.99730444694821611228253978938, −3.01401410519866040547444543657, −2.69116999021782845940189815854, −1.80166700104144915074050944718, −1.77802965902542530348386408135, −0.905341915206997831297072091040, −0.36925402850570488271511580669, 0.36925402850570488271511580669, 0.905341915206997831297072091040, 1.77802965902542530348386408135, 1.80166700104144915074050944718, 2.69116999021782845940189815854, 3.01401410519866040547444543657, 3.99730444694821611228253978938, 4.00134588714583596209304140489, 4.71510774637363447902838894519, 4.99667737141553815695978103341, 5.78491089347295753035898415199, 5.86717995133336860686690317105, 6.59041533299849448563661684845, 6.85115370563623636349413323482, 7.42586523851789983103533892509, 7.65559443042212896723051636877, 8.496691580693985168973122679887, 8.660242259122220289873101619703, 8.788304226857600252919847954097, 9.365106844464828524164077208610

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