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Properties

Label 4-7098e2-1.1-c1e2-0-4
Degree $4$
Conductor $50381604$
Sign $1$
Analytic cond. $3212.37$
Root an. cond. $7.52846$
Motivic weight $1$
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  + 2·2-s + 2·3-s + 3·4-s + 4·6-s − 2·7-s + 4·8-s + 3·9-s − 8·11-s + 6·12-s − 4·14-s + 5·16-s + 6·17-s + 6·18-s − 8·19-s − 4·21-s − 16·22-s + 8·23-s + 8·24-s + 2·25-s + 4·27-s − 6·28-s + 16·29-s + 6·31-s + 6·32-s − 16·33-s + 12·34-s + 9·36-s + ⋯
L(s)  = 1  + 1.41·2-s + 1.15·3-s + 3/2·4-s + 1.63·6-s − 0.755·7-s + 1.41·8-s + 9-s − 2.41·11-s + 1.73·12-s − 1.06·14-s + 5/4·16-s + 1.45·17-s + 1.41·18-s − 1.83·19-s − 0.872·21-s − 3.41·22-s + 1.66·23-s + 1.63·24-s + 2/5·25-s + 0.769·27-s − 1.13·28-s + 2.97·29-s + 1.07·31-s + 1.06·32-s − 2.78·33-s + 2.05·34-s + 3/2·36-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 50381604 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50381604 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(50381604\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2} \cdot 13^{4}\)
Sign: $1$
Analytic conductor: \(3212.37\)
Root analytic conductor: \(7.52846\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 50381604,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(12.01746516\)
\(L(\frac12)\) \(\approx\) \(12.01746516\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{2} \)
3$C_1$ \( ( 1 - T )^{2} \)
7$C_1$ \( ( 1 + T )^{2} \)
13 \( 1 \)
good5$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.5.a_ac
11$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.11.i_bm
17$D_{4}$ \( 1 - 6 T + 31 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.17.ag_bf
19$D_{4}$ \( 1 + 8 T + 42 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.19.i_bq
23$D_{4}$ \( 1 - 8 T + 59 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.23.ai_ch
29$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.29.aq_es
31$D_{4}$ \( 1 - 6 T + 59 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.31.ag_ch
37$D_{4}$ \( 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.37.e_be
41$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \) 2.41.a_bi
43$D_{4}$ \( 1 - 2 T + 75 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.43.ac_cx
47$D_{4}$ \( 1 - 8 T + 98 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.47.ai_du
53$D_{4}$ \( 1 + 20 T + 203 T^{2} + 20 p T^{3} + p^{2} T^{4} \) 2.53.u_hv
59$D_{4}$ \( 1 + 16 T + 179 T^{2} + 16 p T^{3} + p^{2} T^{4} \) 2.59.q_gx
61$C_2^2$ \( 1 + 95 T^{2} + p^{2} T^{4} \) 2.61.a_dr
67$D_{4}$ \( 1 - 4 T + 111 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.67.ae_eh
71$D_{4}$ \( 1 - 10 T + 119 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.71.ak_ep
73$D_{4}$ \( 1 - 8 T + 114 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.73.ai_ek
79$D_{4}$ \( 1 + 12 T + 182 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.79.m_ha
83$C_2^2$ \( 1 + 163 T^{2} + p^{2} T^{4} \) 2.83.a_gh
89$D_{4}$ \( 1 - 20 T + 275 T^{2} - 20 p T^{3} + p^{2} T^{4} \) 2.89.au_kp
97$D_{4}$ \( 1 + 4 T + 90 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.97.e_dm
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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

−7.994068463065389265635760078966, −7.87405973695191219152580869984, −7.29011577315984090902451420925, −7.06860272349254201506043531778, −6.70335194041258897356112218936, −6.31988358257674050945569472898, −5.91775432011937844052370991349, −5.75407998644270619703397290337, −5.03301245263132417125907878165, −4.76493626586257884956253522732, −4.57458174510022555225775822460, −4.43273587248426435368797929675, −3.36618231298388744291594865157, −3.27832777108244891299865378811, −3.12289494237221678317643164818, −2.76368532167070847048574430749, −2.13742124702297399694266540684, −2.13423581081636619030868173097, −1.08395418286688434424738987159, −0.63108740312207414908542673603, 0.63108740312207414908542673603, 1.08395418286688434424738987159, 2.13423581081636619030868173097, 2.13742124702297399694266540684, 2.76368532167070847048574430749, 3.12289494237221678317643164818, 3.27832777108244891299865378811, 3.36618231298388744291594865157, 4.43273587248426435368797929675, 4.57458174510022555225775822460, 4.76493626586257884956253522732, 5.03301245263132417125907878165, 5.75407998644270619703397290337, 5.91775432011937844052370991349, 6.31988358257674050945569472898, 6.70335194041258897356112218936, 7.06860272349254201506043531778, 7.29011577315984090902451420925, 7.87405973695191219152580869984, 7.994068463065389265635760078966

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