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Properties

Label 4-12537-1.1-c1e2-0-0
Degree $4$
Conductor $12537$
Sign $-1$
Analytic cond. $0.799369$
Root an. cond. $0.945555$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 3·4-s + 7-s − 2·9-s + 3·12-s − 13-s + 5·16-s − 8·19-s − 21-s − 8·25-s + 5·27-s − 3·28-s − 11·31-s + 6·36-s − 5·37-s + 39-s + 7·43-s − 5·48-s − 6·49-s + 3·52-s + 8·57-s + 11·61-s − 2·63-s − 3·64-s + 5·67-s + 8·73-s + 8·75-s + ⋯
L(s)  = 1  − 0.577·3-s − 3/2·4-s + 0.377·7-s − 2/3·9-s + 0.866·12-s − 0.277·13-s + 5/4·16-s − 1.83·19-s − 0.218·21-s − 8/5·25-s + 0.962·27-s − 0.566·28-s − 1.97·31-s + 36-s − 0.821·37-s + 0.160·39-s + 1.06·43-s − 0.721·48-s − 6/7·49-s + 0.416·52-s + 1.05·57-s + 1.40·61-s − 0.251·63-s − 3/8·64-s + 0.610·67-s + 0.936·73-s + 0.923·75-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 12537 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12537 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(12537\)    =    \(3^{2} \cdot 7 \cdot 199\)
Sign: $-1$
Analytic conductor: \(0.799369\)
Root analytic conductor: \(0.945555\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 12537,\ (\ :1/2, 1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 $\mathbf{F}_p$
bad3$C_2$ \( 1 + T + p T^{2} \)
7$C_1$$\times$$C_2$ \( ( 1 - T )( 1 + p T^{2} ) \)
199$C_1$$\times$$C_2$ \( ( 1 - T )( 1 + 15 T + p T^{2} ) \)
good2$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.2.a_d
5$C_2^2$ \( 1 + 8 T^{2} + p^{2} T^{4} \) 2.5.a_i
11$C_2^2$ \( 1 - 20 T^{2} + p^{2} T^{4} \) 2.11.a_au
13$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.13.b_u
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.17.a_ac
19$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.19.i_cc
23$C_2^2$ \( 1 + 15 T^{2} + p^{2} T^{4} \) 2.23.a_p
29$C_2^2$ \( 1 + 25 T^{2} + p^{2} T^{4} \) 2.29.a_z
31$C_2$$\times$$C_2$ \( ( 1 + 5 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.31.l_do
37$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.37.f_bm
41$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.41.a_cv
43$C_2$$\times$$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + p T^{2} ) \) 2.43.ah_di
47$C_2^2$ \( 1 - 48 T^{2} + p^{2} T^{4} \) 2.47.a_abw
53$C_2^2$ \( 1 + 65 T^{2} + p^{2} T^{4} \) 2.53.a_cn
59$C_2^2$ \( 1 - 17 T^{2} + p^{2} T^{4} \) 2.59.a_ar
61$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + T + p T^{2} ) \) 2.61.al_eg
67$C_2$$\times$$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.67.af_be
71$C_2^2$ \( 1 + 80 T^{2} + p^{2} T^{4} \) 2.71.a_dc
73$C_2$$\times$$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.73.ai_ej
79$C_2$$\times$$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.79.b_dy
83$C_2^2$ \( 1 - 75 T^{2} + p^{2} T^{4} \) 2.83.a_acx
89$C_2^2$ \( 1 + 108 T^{2} + p^{2} T^{4} \) 2.89.a_ee
97$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.97.c_ek
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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

−10.93847940168557359376355293261, −10.62736370520592967616782055811, −9.795213107434872278172345616262, −9.388807640798493906206967556124, −8.728028446557846585485006860774, −8.371522870542366856046539395925, −7.78712472804842617993544809131, −6.92125038738277596505521512843, −6.11208973522777452262834601500, −5.49357944736289930329450546672, −5.01641343542376593764242257666, −4.19881404080150962529016740442, −3.67164470618081597323630615957, −2.15167302157919725897755561986, 0, 2.15167302157919725897755561986, 3.67164470618081597323630615957, 4.19881404080150962529016740442, 5.01641343542376593764242257666, 5.49357944736289930329450546672, 6.11208973522777452262834601500, 6.92125038738277596505521512843, 7.78712472804842617993544809131, 8.371522870542366856046539395925, 8.728028446557846585485006860774, 9.388807640798493906206967556124, 9.795213107434872278172345616262, 10.62736370520592967616782055811, 10.93847940168557359376355293261

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