Properties

Label 8-384e4-1.1-c3e4-0-7
Degree $8$
Conductor $21743271936$
Sign $1$
Analytic cond. $263505.$
Root an. cond. $4.75990$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 32·7-s − 18·9-s − 72·17-s + 512·23-s + 52·25-s + 160·31-s + 872·41-s + 448·47-s − 316·49-s − 576·63-s + 4.09e3·71-s − 1.32e3·73-s − 992·79-s + 243·81-s − 1.06e3·89-s − 2.44e3·97-s + 3.42e3·103-s + 456·113-s − 2.30e3·119-s − 1.36e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 1.29e3·153-s + ⋯
L(s)  = 1  + 1.72·7-s − 2/3·9-s − 1.02·17-s + 4.64·23-s + 0.415·25-s + 0.926·31-s + 3.32·41-s + 1.39·47-s − 0.921·49-s − 1.15·63-s + 6.84·71-s − 2.11·73-s − 1.41·79-s + 1/3·81-s − 1.26·89-s − 2.55·97-s + 3.27·103-s + 0.379·113-s − 1.77·119-s − 1.02·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.684·153-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{28} \cdot 3^{4}\)
Sign: $1$
Analytic conductor: \(263505.\)
Root analytic conductor: \(4.75990\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{384} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{28} \cdot 3^{4} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )\)

Particular Values

\(L(2)\) \(\approx\) \(7.978498188\)
\(L(\frac12)\) \(\approx\) \(7.978498188\)
\(L(\frac{5}{2})\) 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$C_2$ \( ( 1 + p^{2} T^{2} )^{2} \)
good5$D_4\times C_2$ \( 1 - 52 T^{2} + 18614 T^{4} - 52 p^{6} T^{6} + p^{12} T^{8} \)
7$D_{4}$ \( ( 1 - 16 T + 542 T^{2} - 16 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
11$D_4\times C_2$ \( 1 + 124 p T^{2} + 3795254 T^{4} + 124 p^{7} T^{6} + p^{12} T^{8} \)
13$D_4\times C_2$ \( 1 - 4532 T^{2} + 10475286 T^{4} - 4532 p^{6} T^{6} + p^{12} T^{8} \)
17$D_{4}$ \( ( 1 + 36 T + 6822 T^{2} + 36 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
19$D_4\times C_2$ \( 1 - 11532 T^{2} + 65783830 T^{4} - 11532 p^{6} T^{6} + p^{12} T^{8} \)
23$D_{4}$ \( ( 1 - 256 T + 39886 T^{2} - 256 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
29$D_4\times C_2$ \( 1 - 52308 T^{2} + 1484422166 T^{4} - 52308 p^{6} T^{6} + p^{12} T^{8} \)
31$D_{4}$ \( ( 1 - 80 T + 26030 T^{2} - 80 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
37$D_4\times C_2$ \( 1 - 186708 T^{2} + 13784867446 T^{4} - 186708 p^{6} T^{6} + p^{12} T^{8} \)
41$D_{4}$ \( ( 1 - 436 T + 102166 T^{2} - 436 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
43$D_4\times C_2$ \( 1 - 57900 T^{2} + 11793718966 T^{4} - 57900 p^{6} T^{6} + p^{12} T^{8} \)
47$D_{4}$ \( ( 1 - 224 T + 179422 T^{2} - 224 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
53$D_4\times C_2$ \( 1 - 442740 T^{2} + 87795274358 T^{4} - 442740 p^{6} T^{6} + p^{12} T^{8} \)
59$C_2^2$ \( ( 1 - 305782 T^{2} + p^{6} T^{4} )^{2} \)
61$C_2^2$ \( ( 1 - 348986 T^{2} + p^{6} T^{4} )^{2} \)
67$D_4\times C_2$ \( 1 - 140620 T^{2} + 86210675286 T^{4} - 140620 p^{6} T^{6} + p^{12} T^{8} \)
71$D_{4}$ \( ( 1 - 2048 T + 1763566 T^{2} - 2048 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
73$D_{4}$ \( ( 1 + 660 T + 407702 T^{2} + 660 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
79$D_{4}$ \( ( 1 + 496 T + 323534 T^{2} + 496 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
83$D_4\times C_2$ \( 1 - 1659916 T^{2} + 1244512983830 T^{4} - 1659916 p^{6} T^{6} + p^{12} T^{8} \)
89$D_{4}$ \( ( 1 + 532 T + 1467382 T^{2} + 532 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
97$D_{4}$ \( ( 1 + 1220 T + 586694 T^{2} + 1220 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
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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

−7.948919490273926557684207758723, −7.41307558194206554426927431279, −7.27150192568504865621404658773, −7.05777120539566782326530413174, −6.85387930554797081692058598828, −6.56490473534773537384508999478, −6.26597809018203429468873452697, −6.03312912555073074319391660476, −5.49833039472147976549957587842, −5.47240840413814229215303951401, −4.96531360452056762308303483580, −4.94694672132284673435557998511, −4.82182598867083594077417838878, −4.30823829320853091540522653307, −4.22144921170111837608413568024, −3.71392575480308516797721506940, −3.36036134816789442073500953724, −2.88471657120935676578383145642, −2.72957511084558171217329710992, −2.42482035271289104433297996357, −2.15848546524374831459178904879, −1.32174349401729479741596330884, −1.27693681370273272238060629301, −0.822364584531941637277034558000, −0.49110437182719285386602376106, 0.49110437182719285386602376106, 0.822364584531941637277034558000, 1.27693681370273272238060629301, 1.32174349401729479741596330884, 2.15848546524374831459178904879, 2.42482035271289104433297996357, 2.72957511084558171217329710992, 2.88471657120935676578383145642, 3.36036134816789442073500953724, 3.71392575480308516797721506940, 4.22144921170111837608413568024, 4.30823829320853091540522653307, 4.82182598867083594077417838878, 4.94694672132284673435557998511, 4.96531360452056762308303483580, 5.47240840413814229215303951401, 5.49833039472147976549957587842, 6.03312912555073074319391660476, 6.26597809018203429468873452697, 6.56490473534773537384508999478, 6.85387930554797081692058598828, 7.05777120539566782326530413174, 7.27150192568504865621404658773, 7.41307558194206554426927431279, 7.948919490273926557684207758723

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