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Properties

Label 4-238e2-1.1-c1e2-0-5
Degree $4$
Conductor $56644$
Sign $1$
Analytic cond. $3.61167$
Root an. cond. $1.37856$
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 + 3·4-s − 4·8-s + 6·9-s + 4·13-s + 5·16-s + 2·17-s − 12·18-s + 8·19-s + 10·25-s − 8·26-s − 6·32-s − 4·34-s + 18·36-s − 16·38-s − 8·43-s − 16·47-s − 49-s − 20·50-s + 12·52-s − 20·53-s − 24·59-s + 7·64-s − 8·67-s + 6·68-s − 24·72-s + 24·76-s + ⋯
L(s)  = 1  − 1.41·2-s + 3/2·4-s − 1.41·8-s + 2·9-s + 1.10·13-s + 5/4·16-s + 0.485·17-s − 2.82·18-s + 1.83·19-s + 2·25-s − 1.56·26-s − 1.06·32-s − 0.685·34-s + 3·36-s − 2.59·38-s − 1.21·43-s − 2.33·47-s − 1/7·49-s − 2.82·50-s + 1.66·52-s − 2.74·53-s − 3.12·59-s + 7/8·64-s − 0.977·67-s + 0.727·68-s − 2.82·72-s + 2.75·76-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 56644 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 56644 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(56644\)    =    \(2^{2} \cdot 7^{2} \cdot 17^{2}\)
Sign: $1$
Analytic conductor: \(3.61167\)
Root analytic conductor: \(1.37856\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 56644,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9800067346\)
\(L(\frac12)\) \(\approx\) \(0.9800067346\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{2} \)
7$C_2$ \( 1 + T^{2} \)
17$C_2$ \( 1 - 2 T + p T^{2} \)
good3$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.3.a_ag
5$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.5.a_ak
11$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.11.a_aw
13$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.13.ae_be
19$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.19.ai_cc
23$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \) 2.23.a_s
29$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \) 2.29.a_g
31$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.31.a_c
37$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.37.a_ak
41$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.41.a_as
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.43.i_dy
47$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.47.q_gc
53$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.53.u_hy
59$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.59.y_kc
61$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.61.a_aes
67$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.67.i_fu
71$C_2^2$ \( 1 - 78 T^{2} + p^{2} T^{4} \) 2.71.a_ada
73$C_2^2$ \( 1 - 82 T^{2} + p^{2} T^{4} \) 2.73.a_ade
79$C_2^2$ \( 1 - 94 T^{2} + p^{2} T^{4} \) 2.79.a_adq
83$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.83.i_ha
89$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.89.am_ig
97$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) 2.97.a_afa
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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

−12.35476316039365443178180347356, −11.79363940345498586897158986324, −11.17894939806280487742414525265, −10.85852910727115669316902048057, −10.38557010750087216502196945908, −9.818989973120345306846165255455, −9.570202616221430988824004669051, −9.165773174894196987499953940192, −8.429879563394443700876856131754, −8.022963046821189443495886909603, −7.34604714588017519340733516685, −7.23952794483880087821443207618, −6.34247778860463636263915020989, −6.23364129453637355620609250303, −4.91578518638780977122073699926, −4.74737115838155404846784134701, −3.32199763483264578429423763619, −3.24614941483912415817873523751, −1.52737554662896515067570835275, −1.31393537933617091426787390916, 1.31393537933617091426787390916, 1.52737554662896515067570835275, 3.24614941483912415817873523751, 3.32199763483264578429423763619, 4.74737115838155404846784134701, 4.91578518638780977122073699926, 6.23364129453637355620609250303, 6.34247778860463636263915020989, 7.23952794483880087821443207618, 7.34604714588017519340733516685, 8.022963046821189443495886909603, 8.429879563394443700876856131754, 9.165773174894196987499953940192, 9.570202616221430988824004669051, 9.818989973120345306846165255455, 10.38557010750087216502196945908, 10.85852910727115669316902048057, 11.17894939806280487742414525265, 11.79363940345498586897158986324, 12.35476316039365443178180347356

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