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Properties

Label 4-6272e2-1.1-c1e2-0-6
Degree $4$
Conductor $39337984$
Sign $1$
Analytic cond. $2508.22$
Root an. cond. $7.07687$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s − 2·5-s − 4·11-s − 2·13-s − 4·15-s − 8·17-s + 2·19-s + 4·23-s − 4·25-s − 2·27-s − 12·31-s − 8·33-s + 16·37-s − 4·39-s − 12·43-s − 4·47-s − 16·51-s + 12·53-s + 8·55-s + 4·57-s + 2·59-s − 2·61-s + 4·65-s − 16·67-s + 8·69-s − 4·73-s − 8·75-s + ⋯
L(s)  = 1  + 1.15·3-s − 0.894·5-s − 1.20·11-s − 0.554·13-s − 1.03·15-s − 1.94·17-s + 0.458·19-s + 0.834·23-s − 4/5·25-s − 0.384·27-s − 2.15·31-s − 1.39·33-s + 2.63·37-s − 0.640·39-s − 1.82·43-s − 0.583·47-s − 2.24·51-s + 1.64·53-s + 1.07·55-s + 0.529·57-s + 0.260·59-s − 0.256·61-s + 0.496·65-s − 1.95·67-s + 0.963·69-s − 0.468·73-s − 0.923·75-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 39337984 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39337984 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(39337984\)    =    \(2^{14} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(2508.22\)
Root analytic conductor: \(7.07687\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 39337984,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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)$Isogeny Class over $\mathbf{F}_p$
bad2 \( 1 \)
7 \( 1 \)
good3$D_{4}$ \( 1 - 2 T + 4 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.3.ac_e
5$D_{4}$ \( 1 + 2 T + 8 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.5.c_i
11$D_{4}$ \( 1 + 4 T + 14 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.11.e_o
13$D_{4}$ \( 1 + 2 T + 24 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.13.c_y
17$D_{4}$ \( 1 + 8 T + 38 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.17.i_bm
19$D_{4}$ \( 1 - 2 T + 12 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.19.ac_m
23$D_{4}$ \( 1 - 4 T + 2 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.23.ae_c
29$C_2^2$ \( 1 + 46 T^{2} + p^{2} T^{4} \) 2.29.a_bu
31$D_{4}$ \( 1 + 12 T + 86 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.31.m_di
37$D_{4}$ \( 1 - 16 T + 126 T^{2} - 16 p T^{3} + p^{2} T^{4} \) 2.37.aq_ew
41$C_2^2$ \( 1 + 70 T^{2} + p^{2} T^{4} \) 2.41.a_cs
43$D_{4}$ \( 1 + 12 T + 110 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.43.m_eg
47$D_{4}$ \( 1 + 4 T + 86 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.47.e_di
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.53.am_fm
59$D_{4}$ \( 1 - 2 T + 92 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.59.ac_do
61$D_{4}$ \( 1 + 2 T - 24 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.61.c_ay
67$D_{4}$ \( 1 + 16 T + 150 T^{2} + 16 p T^{3} + p^{2} T^{4} \) 2.67.q_fu
71$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \) 2.71.a_aby
73$D_{4}$ \( 1 + 4 T + 102 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.73.e_dy
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.79.aq_io
83$D_{4}$ \( 1 + 18 T + 244 T^{2} + 18 p T^{3} + p^{2} T^{4} \) 2.83.s_jk
89$D_{4}$ \( 1 - 4 T + 134 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.89.ae_fe
97$D_{4}$ \( 1 + 8 T + 198 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.97.i_hq
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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

−7.948818919488398160726333560706, −7.43998876743000581463994358104, −7.31003724112895062071724493660, −7.16835962094261489841952232759, −6.36759962798243432206291716335, −6.28441853973913646301673749067, −5.56734612949879290319990936268, −5.44208592528939446961605697305, −4.76934367674532220389670324813, −4.65992190911625580120420154320, −4.14847408448494649388695057910, −3.80926768636345113992586745861, −3.16696771579471733991759654098, −3.15306360203289288216961820494, −2.53330423651033050960874388709, −2.19294115135386060735377097562, −1.91186885221378642424102552386, −0.999322108200962951112315387171, 0, 0, 0.999322108200962951112315387171, 1.91186885221378642424102552386, 2.19294115135386060735377097562, 2.53330423651033050960874388709, 3.15306360203289288216961820494, 3.16696771579471733991759654098, 3.80926768636345113992586745861, 4.14847408448494649388695057910, 4.65992190911625580120420154320, 4.76934367674532220389670324813, 5.44208592528939446961605697305, 5.56734612949879290319990936268, 6.28441853973913646301673749067, 6.36759962798243432206291716335, 7.16835962094261489841952232759, 7.31003724112895062071724493660, 7.43998876743000581463994358104, 7.948818919488398160726333560706

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