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Properties

Label 4-792e2-1.1-c1e2-0-26
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-s − 4-s − 3·8-s + 4·11-s − 16-s − 8·17-s + 4·22-s + 10·25-s − 4·29-s − 8·31-s + 5·32-s − 8·34-s + 16·37-s + 8·41-s − 4·44-s + 2·49-s + 10·50-s − 4·58-s − 8·62-s + 7·64-s + 8·68-s + 16·74-s + 8·82-s − 8·83-s − 12·88-s − 4·97-s + 2·98-s + ⋯
L(s)  = 1  + 0.707·2-s − 1/2·4-s − 1.06·8-s + 1.20·11-s − 1/4·16-s − 1.94·17-s + 0.852·22-s + 2·25-s − 0.742·29-s − 1.43·31-s + 0.883·32-s − 1.37·34-s + 2.63·37-s + 1.24·41-s − 0.603·44-s + 2/7·49-s + 1.41·50-s − 0.525·58-s − 1.01·62-s + 7/8·64-s + 0.970·68-s + 1.85·74-s + 0.883·82-s − 0.878·83-s − 1.27·88-s − 0.406·97-s + 0.202·98-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

\(L(1)\) \(\approx\) \(2.058299275\)
\(L(\frac12)\) \(\approx\) \(2.058299275\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - T + p T^{2} \)
3 \( 1 \)
11$C_2$ \( 1 - 4 T + p T^{2} \)
good5$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.5.a_ak
7$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.7.a_ac
13$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \) 2.13.a_ag
17$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.17.i_bu
19$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \) 2.19.a_ag
23$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.23.a_c
29$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.29.e_ac
31$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.31.i_ck
37$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) 2.37.aq_fe
41$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.41.ai_dq
43$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.43.a_aw
47$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \) 2.47.a_ao
53$C_2^2$ \( 1 + 86 T^{2} + p^{2} T^{4} \) 2.53.a_di
59$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \) 2.59.a_bm
61$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \) 2.61.a_ba
67$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.67.a_eo
71$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.71.a_c
73$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.73.a_bu
79$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \) 2.79.a_as
83$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.83.i_eo
89$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \) 2.89.a_by
97$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.97.e_hq
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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

−8.671460329837372450209993645902, −7.88390591795779701656894140322, −7.45509829577586183380880834585, −6.85858688864893954498436770232, −6.42910150387716351484663436476, −6.23395565084076397021939404120, −5.48729423501560285109521771709, −5.15761501207680469210158388583, −4.41855807939959967222179346040, −4.15780525881864344441323000860, −3.89407467832143456513551853660, −2.91160227261098715446466119546, −2.62479949450785067792117952970, −1.65257315265339974257143638207, −0.66782208448803880492810804458, 0.66782208448803880492810804458, 1.65257315265339974257143638207, 2.62479949450785067792117952970, 2.91160227261098715446466119546, 3.89407467832143456513551853660, 4.15780525881864344441323000860, 4.41855807939959967222179346040, 5.15761501207680469210158388583, 5.48729423501560285109521771709, 6.23395565084076397021939404120, 6.42910150387716351484663436476, 6.85858688864893954498436770232, 7.45509829577586183380880834585, 7.88390591795779701656894140322, 8.671460329837372450209993645902

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