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Properties

Label 4-792e2-1.1-c1e2-0-1
Degree $4$
Conductor $627264$
Sign $1$
Analytic cond. $39.9948$
Root an. cond. $2.51478$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s − 3-s + 2·4-s + 2·6-s − 2·7-s + 9-s + 2·11-s − 2·12-s − 13-s + 4·14-s − 4·16-s − 2·18-s + 2·21-s − 4·22-s + 25-s + 2·26-s − 27-s − 4·28-s − 12·29-s + 8·32-s − 2·33-s + 2·36-s + 39-s − 4·42-s + 4·44-s + 4·48-s − 11·49-s + ⋯
L(s)  = 1  − 1.41·2-s − 0.577·3-s + 4-s + 0.816·6-s − 0.755·7-s + 1/3·9-s + 0.603·11-s − 0.577·12-s − 0.277·13-s + 1.06·14-s − 16-s − 0.471·18-s + 0.436·21-s − 0.852·22-s + 1/5·25-s + 0.392·26-s − 0.192·27-s − 0.755·28-s − 2.22·29-s + 1.41·32-s − 0.348·33-s + 1/3·36-s + 0.160·39-s − 0.617·42-s + 0.603·44-s + 0.577·48-s − 1.57·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 627264 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 627264 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(627264\)    =    \(2^{6} \cdot 3^{4} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(39.9948\)
Root analytic conductor: \(2.51478\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 627264,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3641924696\)
\(L(\frac12)\) \(\approx\) \(0.3641924696\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + p T + p T^{2} \)
3$C_1$ \( 1 + T \)
11$C_2$ \( 1 - 2 T + p T^{2} \)
good5$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \) 2.5.a_ab
7$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.7.c_p
13$C_2$$\times$$C_2$ \( ( 1 - T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.13.b_y
17$C_2^2$ \( 1 - 25 T^{2} + p^{2} T^{4} \) 2.17.a_az
19$C_2^2$ \( 1 - 7 T^{2} + p^{2} T^{4} \) 2.19.a_ah
23$C_2^2$ \( 1 + 33 T^{2} + p^{2} T^{4} \) 2.23.a_bh
29$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.29.m_dq
31$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.31.a_c
37$C_2^2$ \( 1 + 41 T^{2} + p^{2} T^{4} \) 2.37.a_bp
41$C_2^2$ \( 1 + 35 T^{2} + p^{2} T^{4} \) 2.41.a_bj
43$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \) 2.43.a_acs
47$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \) 2.47.a_s
53$C_2^2$ \( 1 + 11 T^{2} + p^{2} T^{4} \) 2.53.a_l
59$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.59.e_di
61$C_2$$\times$$C_2$ \( ( 1 - 15 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.61.aj_bg
67$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) 2.67.ap_go
71$C_2^2$ \( 1 + 74 T^{2} + p^{2} T^{4} \) 2.71.a_cw
73$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.73.a_dt
79$C_2$$\times$$C_2$ \( ( 1 + 13 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.79.bd_oc
83$C_2^2$ \( 1 - 67 T^{2} + p^{2} T^{4} \) 2.83.a_acp
89$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.89.k_es
97$C_2$$\times$$C_2$ \( ( 1 - 17 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.97.at_iu
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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

−8.597535703688339175884596876897, −7.941131356787931938528355545742, −7.49859374028433657269994503906, −7.03873207586578073963834410385, −6.84952219408898290749391909270, −6.18903728282673169821278407403, −5.85547369369790703747762555916, −5.21278839975729832760233125210, −4.69523767414760788827716693787, −4.02973036183346233377902232276, −3.57586812602217524487867035838, −2.81899966858257210535859126488, −1.99136293779289589858748837152, −1.43699992124098362387064357826, −0.41161964303919691304925155264, 0.41161964303919691304925155264, 1.43699992124098362387064357826, 1.99136293779289589858748837152, 2.81899966858257210535859126488, 3.57586812602217524487867035838, 4.02973036183346233377902232276, 4.69523767414760788827716693787, 5.21278839975729832760233125210, 5.85547369369790703747762555916, 6.18903728282673169821278407403, 6.84952219408898290749391909270, 7.03873207586578073963834410385, 7.49859374028433657269994503906, 7.941131356787931938528355545742, 8.597535703688339175884596876897

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