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Properties

Label 4-238e2-1.1-c1e2-0-2
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-s + 2·3-s − 3·5-s + 2·6-s − 7-s − 8-s + 3·9-s − 3·10-s + 4·13-s − 14-s − 6·15-s − 16-s + 17-s + 3·18-s + 7·19-s − 2·21-s + 3·23-s − 2·24-s + 5·25-s + 4·26-s + 10·27-s − 12·29-s − 6·30-s + 4·31-s + 34-s + 3·35-s + 7·37-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.15·3-s − 1.34·5-s + 0.816·6-s − 0.377·7-s − 0.353·8-s + 9-s − 0.948·10-s + 1.10·13-s − 0.267·14-s − 1.54·15-s − 1/4·16-s + 0.242·17-s + 0.707·18-s + 1.60·19-s − 0.436·21-s + 0.625·23-s − 0.408·24-s + 25-s + 0.784·26-s + 1.92·27-s − 2.22·29-s − 1.09·30-s + 0.718·31-s + 0.171·34-s + 0.507·35-s + 1.15·37-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

\(L(1)\) \(\approx\) \(2.311845221\)
\(L(\frac12)\) \(\approx\) \(2.311845221\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 - 2 T + T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.3.ac_b
5$C_2^2$ \( 1 + 3 T + 4 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.5.d_e
11$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.11.a_al
13$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.13.ae_be
19$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + T + p T^{2} ) \) 2.19.ah_be
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$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.31.ae_ap
37$C_2^2$ \( 1 - 7 T + 12 T^{2} - 7 p T^{3} + p^{2} T^{4} \) 2.37.ah_m
41$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.41.am_eo
43$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.43.c_dj
47$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.47.a_abv
53$C_2^2$ \( 1 - 6 T - 17 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.53.ag_ar
59$C_2^2$ \( 1 + 9 T + 22 T^{2} + 9 p T^{3} + p^{2} T^{4} \) 2.59.j_w
61$C_2$ \( ( 1 + T + p T^{2} )( 1 + 13 T + p T^{2} ) \) 2.61.o_ff
67$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.67.f_abq
71$C_2$ \( ( 1 + 15 T + p T^{2} )^{2} \) 2.71.be_od
73$C_2^2$ \( 1 + 2 T - 69 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.73.c_acr
79$C_2^2$ \( 1 - 16 T + 177 T^{2} - 16 p T^{3} + p^{2} T^{4} \) 2.79.aq_gv
83$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.83.y_ly
89$C_2^2$ \( 1 + 9 T - 8 T^{2} + 9 p T^{3} + p^{2} T^{4} \) 2.89.j_ai
97$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \) 2.97.abc_pa
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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

−12.38700648033543594728100825669, −12.08353889485174610087518105009, −11.40330382380248736658756278003, −11.17890559002194399568543280177, −10.56445841124125399109608242166, −9.831937047324308557701770563747, −9.411419025991763843480253770364, −8.802079754560968294820063470568, −8.638411933679671803457793136768, −7.79630178006575650532045099138, −7.35463223906308476682206953072, −7.31460545929422226349718416119, −6.00822455021959700007313727541, −5.96754541743606809381183389535, −4.58993442494709825612973642853, −4.58299996748710923049620542147, −3.49305445327219861685086660837, −3.41467032565540639036279312810, −2.73967128700619818056240944954, −1.21846385450604278795031072134, 1.21846385450604278795031072134, 2.73967128700619818056240944954, 3.41467032565540639036279312810, 3.49305445327219861685086660837, 4.58299996748710923049620542147, 4.58993442494709825612973642853, 5.96754541743606809381183389535, 6.00822455021959700007313727541, 7.31460545929422226349718416119, 7.35463223906308476682206953072, 7.79630178006575650532045099138, 8.638411933679671803457793136768, 8.802079754560968294820063470568, 9.411419025991763843480253770364, 9.831937047324308557701770563747, 10.56445841124125399109608242166, 11.17890559002194399568543280177, 11.40330382380248736658756278003, 12.08353889485174610087518105009, 12.38700648033543594728100825669

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