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Properties

Label 4-238e2-1.1-c1e2-0-6
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 + 2·3-s + 3·4-s + 2·5-s − 4·6-s − 2·7-s − 4·8-s + 2·9-s − 4·10-s + 6·11-s + 6·12-s + 4·13-s + 4·14-s + 4·15-s + 5·16-s + 2·17-s − 4·18-s − 8·19-s + 6·20-s − 4·21-s − 12·22-s + 16·23-s − 8·24-s − 2·25-s − 8·26-s + 6·27-s − 6·28-s + ⋯
L(s)  = 1  − 1.41·2-s + 1.15·3-s + 3/2·4-s + 0.894·5-s − 1.63·6-s − 0.755·7-s − 1.41·8-s + 2/3·9-s − 1.26·10-s + 1.80·11-s + 1.73·12-s + 1.10·13-s + 1.06·14-s + 1.03·15-s + 5/4·16-s + 0.485·17-s − 0.942·18-s − 1.83·19-s + 1.34·20-s − 0.872·21-s − 2.55·22-s + 3.33·23-s − 1.63·24-s − 2/5·25-s − 1.56·26-s + 1.15·27-s − 1.13·28-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

\(L(1)\) \(\approx\) \(1.357273275\)
\(L(\frac12)\) \(\approx\) \(1.357273275\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{2} \)
17$C_1$ \( ( 1 - T )^{2} \)
good3$C_2^2$ \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.3.ac_c
5$D_{4}$ \( 1 - 2 T + 6 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.5.ac_g
11$C_4$ \( 1 - 6 T + 26 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.11.ag_ba
13$D_{4}$ \( 1 - 4 T + 10 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.13.ae_k
19$C_4$ \( 1 + 8 T + 34 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.19.i_bi
23$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.23.aq_eg
29$D_{4}$ \( 1 - 2 T + 14 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.29.ac_o
31$D_{4}$ \( 1 + 12 T + 78 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.31.m_da
37$D_{4}$ \( 1 + 2 T + 70 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.37.c_cs
41$C_4$ \( 1 + 16 T + 126 T^{2} + 16 p T^{3} + p^{2} T^{4} \) 2.41.q_ew
43$D_{4}$ \( 1 + 4 T + 70 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.43.e_cs
47$D_{4}$ \( 1 - 4 T + 78 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.47.ae_da
53$C_2^2$ \( 1 + 86 T^{2} + p^{2} T^{4} \) 2.53.a_di
59$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.59.m_fy
61$D_{4}$ \( 1 + 2 T + 118 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.61.c_eo
67$D_{4}$ \( 1 + 12 T + 150 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.67.m_fu
71$C_4$ \( 1 - 4 T + 126 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.71.ae_ew
73$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \) 2.73.a_abi
79$D_{4}$ \( 1 + 12 T + 174 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.79.m_gs
83$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.83.ae_go
89$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.89.ae_ha
97$D_{4}$ \( 1 + 8 T + 190 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.97.i_hi
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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

−12.28832103067981868195069364999, −11.92612142471837471706435068419, −10.98017512674928301187353752954, −10.90494291763676779852377629713, −10.36006400201378464173297109921, −9.752887239297725819796461544628, −9.173706923637329481742301738429, −9.103938484655653492425357032401, −8.610264838777983408651887538747, −8.451991930124608559593159911555, −7.41093439925521310147741227555, −6.83921001446596737633967757560, −6.66062702753093859465403758477, −6.11347826531456823537239071134, −5.31932687459869695779278148644, −4.23178491420962269647117850282, −3.28061967569918783190091349429, −3.13391409231110049395176238944, −1.87504138844501808137256975525, −1.36349622199492200919804644331, 1.36349622199492200919804644331, 1.87504138844501808137256975525, 3.13391409231110049395176238944, 3.28061967569918783190091349429, 4.23178491420962269647117850282, 5.31932687459869695779278148644, 6.11347826531456823537239071134, 6.66062702753093859465403758477, 6.83921001446596737633967757560, 7.41093439925521310147741227555, 8.451991930124608559593159911555, 8.610264838777983408651887538747, 9.103938484655653492425357032401, 9.173706923637329481742301738429, 9.752887239297725819796461544628, 10.36006400201378464173297109921, 10.90494291763676779852377629713, 10.98017512674928301187353752954, 11.92612142471837471706435068419, 12.28832103067981868195069364999

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