Properties

Label 4-2738e2-1.1-c1e2-0-7
Degree $4$
Conductor $7496644$
Sign $1$
Analytic cond. $477.992$
Root an. cond. $4.67579$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s − 2·3-s + 3·4-s − 4·6-s + 8·7-s + 4·8-s − 6·11-s − 6·12-s − 12·13-s + 16·14-s + 5·16-s − 6·19-s − 16·21-s − 12·22-s − 6·23-s − 8·24-s − 7·25-s − 24·26-s + 2·27-s + 24·28-s − 12·29-s − 6·31-s + 6·32-s + 12·33-s − 12·38-s + 24·39-s + 6·41-s + ⋯
L(s)  = 1  + 1.41·2-s − 1.15·3-s + 3/2·4-s − 1.63·6-s + 3.02·7-s + 1.41·8-s − 1.80·11-s − 1.73·12-s − 3.32·13-s + 4.27·14-s + 5/4·16-s − 1.37·19-s − 3.49·21-s − 2.55·22-s − 1.25·23-s − 1.63·24-s − 7/5·25-s − 4.70·26-s + 0.384·27-s + 4.53·28-s − 2.22·29-s − 1.07·31-s + 1.06·32-s + 2.08·33-s − 1.94·38-s + 3.84·39-s + 0.937·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7496644 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7496644 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(7496644\)    =    \(2^{2} \cdot 37^{4}\)
Sign: $1$
Analytic conductor: \(477.992\)
Root analytic conductor: \(4.67579\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 7496644,\ (\ :1/2, 1/2),\ 1)\)

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$C_1$ \( ( 1 - T )^{2} \)
37 \( 1 \)
good3$D_{4}$ \( 1 + 2 T + 4 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.3.c_e
5$C_2^2$ \( 1 + 7 T^{2} + p^{2} T^{4} \) 2.5.a_h
7$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.7.ai_be
11$D_{4}$ \( 1 + 6 T + 28 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.11.g_bc
13$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.13.m_ck
17$C_2^2$ \( 1 + 31 T^{2} + p^{2} T^{4} \) 2.17.a_bf
19$D_{4}$ \( 1 + 6 T + 44 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.19.g_bs
23$D_{4}$ \( 1 + 6 T + 52 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.23.g_ca
29$D_{4}$ \( 1 + 12 T + 91 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.29.m_dn
31$D_{4}$ \( 1 + 6 T + 68 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.31.g_cq
41$D_{4}$ \( 1 - 6 T + 79 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.41.ag_db
43$D_{4}$ \( 1 + 12 T + 110 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.43.m_eg
47$D_{4}$ \( 1 - 6 T + 28 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.47.ag_bc
53$D_{4}$ \( 1 + 12 T + 130 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.53.m_fa
59$D_{4}$ \( 1 - 12 T + 142 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.59.am_fm
61$C_2^2$ \( 1 + 12 T + 83 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.61.m_df
67$D_{4}$ \( 1 + 2 T + 108 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.67.c_ee
71$D_{4}$ \( 1 - 12 T + 166 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.71.am_gk
73$D_{4}$ \( 1 - 4 T + 42 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.73.ae_bq
79$D_{4}$ \( 1 + 6 T + 140 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.79.g_fk
83$D_{4}$ \( 1 - 6 T + 100 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.83.ag_dw
89$C_2^2$ \( 1 + 151 T^{2} + p^{2} T^{4} \) 2.89.a_fv
97$C_2^2$ \( 1 + 167 T^{2} + p^{2} T^{4} \) 2.97.a_gl
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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

−8.180071207794326806558838194665, −8.012173746432723950033977242993, −7.83184682319298438818118189175, −7.46553726416567268523156837988, −7.25071350682856862054070328328, −6.69993182697855990265329418484, −5.82071839124631898483885496455, −5.81289369038113307435763786778, −5.35956151270655253026908501179, −5.12976953069755643280195062952, −4.76604345504227917738043082450, −4.66011039835527908854368369970, −4.11024311746952650002856422775, −3.62636101916346117908130192803, −2.63437081315533679144183692563, −2.26962614827884180643976016929, −1.98297860857396679666937430012, −1.76028871095411217756787445199, 0, 0, 1.76028871095411217756787445199, 1.98297860857396679666937430012, 2.26962614827884180643976016929, 2.63437081315533679144183692563, 3.62636101916346117908130192803, 4.11024311746952650002856422775, 4.66011039835527908854368369970, 4.76604345504227917738043082450, 5.12976953069755643280195062952, 5.35956151270655253026908501179, 5.81289369038113307435763786778, 5.82071839124631898483885496455, 6.69993182697855990265329418484, 7.25071350682856862054070328328, 7.46553726416567268523156837988, 7.83184682319298438818118189175, 8.012173746432723950033977242993, 8.180071207794326806558838194665

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