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Properties

Label 4-792e2-1.1-c1e2-0-15
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 + 9-s − 5·11-s − 2·12-s − 4·16-s + 4·17-s + 2·18-s − 7·19-s − 10·22-s + 4·25-s − 27-s − 8·32-s + 5·33-s + 8·34-s + 2·36-s − 14·38-s + 4·41-s + 11·43-s − 10·44-s + 4·48-s − 8·49-s + 8·50-s − 4·51-s − 2·54-s + ⋯
L(s)  = 1  + 1.41·2-s − 0.577·3-s + 4-s − 0.816·6-s + 1/3·9-s − 1.50·11-s − 0.577·12-s − 16-s + 0.970·17-s + 0.471·18-s − 1.60·19-s − 2.13·22-s + 4/5·25-s − 0.192·27-s − 1.41·32-s + 0.870·33-s + 1.37·34-s + 1/3·36-s − 2.27·38-s + 0.624·41-s + 1.67·43-s − 1.50·44-s + 0.577·48-s − 8/7·49-s + 1.13·50-s − 0.560·51-s − 0.272·54-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

\(L(1)\) \(\approx\) \(2.373471874\)
\(L(\frac12)\) \(\approx\) \(2.373471874\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + 5 T + p T^{2} \)
good5$C_2^2$ \( 1 - 4 T^{2} + p^{2} T^{4} \) 2.5.a_ae
7$C_2^2$ \( 1 + 8 T^{2} + p^{2} T^{4} \) 2.7.a_i
13$C_2^2$ \( 1 + 8 T^{2} + p^{2} T^{4} \) 2.13.a_i
17$C_2$$\times$$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.17.ae_n
19$C_2$$\times$$C_2$ \( ( 1 + 3 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.19.h_by
23$C_2^2$ \( 1 - 20 T^{2} + p^{2} T^{4} \) 2.23.a_au
29$C_2^2$ \( 1 - 35 T^{2} + p^{2} T^{4} \) 2.29.a_abj
31$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.31.a_cb
37$C_2^2$ \( 1 + T^{2} + p^{2} T^{4} \) 2.37.a_b
41$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.41.ae_ao
43$C_2$$\times$$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) 2.43.al_ek
47$C_2^2$ \( 1 - 20 T^{2} + p^{2} T^{4} \) 2.47.a_au
53$C_2^2$ \( 1 + 32 T^{2} + p^{2} T^{4} \) 2.53.a_bg
59$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.59.f_dq
61$C_2^2$ \( 1 - 4 T^{2} + p^{2} T^{4} \) 2.61.a_ae
67$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.67.ab_da
71$C_2^2$ \( 1 + 8 T^{2} + p^{2} T^{4} \) 2.71.a_i
73$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + T + p T^{2} ) \) 2.73.ad_fm
79$C_2^2$ \( 1 - 68 T^{2} + p^{2} T^{4} \) 2.79.a_acq
83$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) 2.83.aq_ig
89$C_2$$\times$$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 - T + p T^{2} ) \) 2.89.ar_hm
97$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) 2.97.aq_ji
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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

−8.213681228634129150628772367080, −7.78404811806858777341251508851, −7.45514517564893129369499025428, −6.83459959766881535923178655356, −6.28732434370147294161226601665, −6.06124091684267591094526776338, −5.55744382668610693833944584042, −5.04201910014136492908927863462, −4.75297245108940322975403997533, −4.28210186770296526522785637160, −3.63053394637834921523895927674, −3.10778028111734363044953665165, −2.49547805093570700068161992494, −1.96385902969954101224519807532, −0.61666213589490070667326346162, 0.61666213589490070667326346162, 1.96385902969954101224519807532, 2.49547805093570700068161992494, 3.10778028111734363044953665165, 3.63053394637834921523895927674, 4.28210186770296526522785637160, 4.75297245108940322975403997533, 5.04201910014136492908927863462, 5.55744382668610693833944584042, 6.06124091684267591094526776338, 6.28732434370147294161226601665, 6.83459959766881535923178655356, 7.45514517564893129369499025428, 7.78404811806858777341251508851, 8.213681228634129150628772367080

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