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Properties

Label 4-238e2-1.1-c1e2-0-4
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

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 5-s − 7-s + 8-s + 3·9-s − 10-s + 6·11-s + 8·13-s + 14-s − 16-s − 17-s − 3·18-s − 19-s − 6·22-s + 3·23-s + 5·25-s − 8·26-s − 12·29-s + 34-s − 35-s + 3·37-s + 38-s + 40-s − 24·41-s + 6·43-s + 3·45-s − 3·46-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.447·5-s − 0.377·7-s + 0.353·8-s + 9-s − 0.316·10-s + 1.80·11-s + 2.21·13-s + 0.267·14-s − 1/4·16-s − 0.242·17-s − 0.707·18-s − 0.229·19-s − 1.27·22-s + 0.625·23-s + 25-s − 1.56·26-s − 2.22·29-s + 0.171·34-s − 0.169·35-s + 0.493·37-s + 0.162·38-s + 0.158·40-s − 3.74·41-s + 0.914·43-s + 0.447·45-s − 0.442·46-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

\(L(1)\) \(\approx\) \(1.259056798\)
\(L(\frac12)\) \(\approx\) \(1.259056798\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T + T^{2} \)
7$C_2$ \( 1 + T + p T^{2} \)
17$C_2$ \( 1 + T + T^{2} \)
good3$C_2$ \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) 2.3.a_ad
5$C_2^2$ \( 1 - T - 4 T^{2} - p T^{3} + p^{2} T^{4} \) 2.5.ab_ae
11$C_2^2$ \( 1 - 6 T + 25 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.11.ag_z
13$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.13.ai_bq
19$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.19.b_as
23$C_2^2$ \( 1 - 3 T - 14 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.23.ad_ao
29$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.29.m_dq
31$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.31.a_abf
37$C_2^2$ \( 1 - 3 T - 28 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.37.ad_abc
41$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.41.y_is
43$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.43.ag_dr
47$C_2^2$ \( 1 - 10 T + 53 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.47.ak_cb
53$C_2^2$ \( 1 - 2 T - 49 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.53.ac_abx
59$C_2^2$ \( 1 + 13 T + 110 T^{2} + 13 p T^{3} + p^{2} T^{4} \) 2.59.n_eg
61$C_2^2$ \( 1 - 10 T + 39 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.61.ak_bn
67$C_2^2$ \( 1 - 3 T - 58 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.67.ad_acg
71$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.71.g_fv
73$C_2^2$ \( 1 - 4 T - 57 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.73.ae_acf
79$C_2^2$ \( 1 + 16 T + 177 T^{2} + 16 p T^{3} + p^{2} T^{4} \) 2.79.q_gv
83$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.83.y_ly
89$C_2^2$ \( 1 + T - 88 T^{2} + p T^{3} + p^{2} T^{4} \) 2.89.b_adk
97$C_2$ \( ( 1 - 16 T + p T^{2} )^{2} \) 2.97.abg_ri
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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

−12.43820927935911293414446924503, −11.66746070004744818660713275421, −11.41045644772336745742136575460, −10.86967899043113774566720831597, −10.41248196193043905699795779046, −9.947677610799746911912556191877, −9.246115960763628558188974855522, −9.147136579098406572994837023750, −8.622819279139183188990598666156, −8.231247611676133100326390591181, −7.11144324415998875056428642835, −7.07593960886437759255678562321, −6.37334428593402730669000401208, −5.97200700882171674282260106669, −5.20058952545748030234893037598, −4.22957758487800151696627993962, −3.87204460522205565338096139841, −3.22790607244797484444602152852, −1.67171097423438779715177038060, −1.30986922391875421353848968600, 1.30986922391875421353848968600, 1.67171097423438779715177038060, 3.22790607244797484444602152852, 3.87204460522205565338096139841, 4.22957758487800151696627993962, 5.20058952545748030234893037598, 5.97200700882171674282260106669, 6.37334428593402730669000401208, 7.07593960886437759255678562321, 7.11144324415998875056428642835, 8.231247611676133100326390591181, 8.622819279139183188990598666156, 9.147136579098406572994837023750, 9.246115960763628558188974855522, 9.947677610799746911912556191877, 10.41248196193043905699795779046, 10.86967899043113774566720831597, 11.41045644772336745742136575460, 11.66746070004744818660713275421, 12.43820927935911293414446924503

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