Properties

Label 4-2738e2-1.1-c1e2-0-2
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 $0$

Origins

Origins of factors

Downloads

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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: \(0\)
Selberg data: \((4,\ 7496644,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.834826282\)
\(L(\frac12)\) \(\approx\) \(1.834826282\)
\(L(\frac{3}{2})\) 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.am_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.ag_bs
23$D_{4}$ \( 1 - 6 T + 52 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.23.ag_ca
29$D_{4}$ \( 1 - 12 T + 91 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.29.am_dn
31$D_{4}$ \( 1 - 6 T + 68 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.31.ag_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.am_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.m_fm
61$C_2^2$ \( 1 - 12 T + 83 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.61.am_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.ag_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

−8.773951100717292849139924840591, −8.599488106621334819476138391860, −8.032916028100805766011187471310, −8.022600604152041363252689207165, −7.82977531527585309538076604026, −7.45761164871895140602240332432, −6.55473864654754237781971426335, −6.49846060521260869608387767596, −5.80110487608468610912328065791, −5.66918237008087793490332094782, −5.21737129514302875381338256246, −5.00043054811138623370139865911, −4.36253531476792157809360052384, −3.96988743623596276111019634378, −3.13037782551287186079151416117, −2.71225044833791298079279909190, −2.10434669388422248628612032688, −1.41394119797288541573371244453, −1.00284266146855375015938965620, −0.834346030079800130478197683735, 0.834346030079800130478197683735, 1.00284266146855375015938965620, 1.41394119797288541573371244453, 2.10434669388422248628612032688, 2.71225044833791298079279909190, 3.13037782551287186079151416117, 3.96988743623596276111019634378, 4.36253531476792157809360052384, 5.00043054811138623370139865911, 5.21737129514302875381338256246, 5.66918237008087793490332094782, 5.80110487608468610912328065791, 6.49846060521260869608387767596, 6.55473864654754237781971426335, 7.45761164871895140602240332432, 7.82977531527585309538076604026, 8.022600604152041363252689207165, 8.032916028100805766011187471310, 8.599488106621334819476138391860, 8.773951100717292849139924840591

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