Properties

Label 2-296208-1.1-c1-0-120
Degree $2$
Conductor $296208$
Sign $-1$
Analytic cond. $2365.23$
Root an. cond. $48.6336$
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  + 2·5-s − 2·7-s + 7·13-s − 17-s + 3·19-s − 6·23-s − 25-s − 2·29-s − 2·31-s − 4·35-s − 8·37-s + 2·41-s + 9·43-s − 7·47-s − 3·49-s + 10·53-s − 12·59-s + 12·61-s + 14·65-s − 5·67-s − 6·71-s + 6·73-s + 4·79-s + 7·83-s − 2·85-s + 3·89-s − 14·91-s + ⋯
L(s)  = 1  + 0.894·5-s − 0.755·7-s + 1.94·13-s − 0.242·17-s + 0.688·19-s − 1.25·23-s − 1/5·25-s − 0.371·29-s − 0.359·31-s − 0.676·35-s − 1.31·37-s + 0.312·41-s + 1.37·43-s − 1.02·47-s − 3/7·49-s + 1.37·53-s − 1.56·59-s + 1.53·61-s + 1.73·65-s − 0.610·67-s − 0.712·71-s + 0.702·73-s + 0.450·79-s + 0.768·83-s − 0.216·85-s + 0.317·89-s − 1.46·91-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 296208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 296208 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(296208\)    =    \(2^{4} \cdot 3^{2} \cdot 11^{2} \cdot 17\)
Sign: $-1$
Analytic conductor: \(2365.23\)
Root analytic conductor: \(48.6336\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 296208,\ (\ :1/2),\ -1)\)

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$$F_p(T)$Isogeny Class over $\mathbf{F}_p$
bad2 \( 1 \)
3 \( 1 \)
11 \( 1 \)
17 \( 1 + T \)
good5 \( 1 - 2 T + p T^{2} \) 1.5.ac
7 \( 1 + 2 T + p T^{2} \) 1.7.c
13 \( 1 - 7 T + p T^{2} \) 1.13.ah
19 \( 1 - 3 T + p T^{2} \) 1.19.ad
23 \( 1 + 6 T + p T^{2} \) 1.23.g
29 \( 1 + 2 T + p T^{2} \) 1.29.c
31 \( 1 + 2 T + p T^{2} \) 1.31.c
37 \( 1 + 8 T + p T^{2} \) 1.37.i
41 \( 1 - 2 T + p T^{2} \) 1.41.ac
43 \( 1 - 9 T + p T^{2} \) 1.43.aj
47 \( 1 + 7 T + p T^{2} \) 1.47.h
53 \( 1 - 10 T + p T^{2} \) 1.53.ak
59 \( 1 + 12 T + p T^{2} \) 1.59.m
61 \( 1 - 12 T + p T^{2} \) 1.61.am
67 \( 1 + 5 T + p T^{2} \) 1.67.f
71 \( 1 + 6 T + p T^{2} \) 1.71.g
73 \( 1 - 6 T + p T^{2} \) 1.73.ag
79 \( 1 - 4 T + p T^{2} \) 1.79.ae
83 \( 1 - 7 T + p T^{2} \) 1.83.ah
89 \( 1 - 3 T + p T^{2} \) 1.89.ad
97 \( 1 - 8 T + p T^{2} \) 1.97.ai
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−13.05728695053611, −12.52595148428033, −12.06430728593004, −11.50082175307439, −11.06608610586963, −10.60345412111397, −10.12080120237044, −9.767741663254332, −9.192904953214458, −8.932799826554965, −8.362558465796883, −7.868720572622599, −7.282396304205192, −6.734236620914425, −6.125191405379149, −6.021309369916319, −5.545906040160762, −4.926614813286769, −4.175929843660784, −3.552831113005120, −3.486442858042052, −2.591373066820552, −1.996132499099126, −1.516921356038569, −0.8485420089915492, 0, 0.8485420089915492, 1.516921356038569, 1.996132499099126, 2.591373066820552, 3.486442858042052, 3.552831113005120, 4.175929843660784, 4.926614813286769, 5.545906040160762, 6.021309369916319, 6.125191405379149, 6.734236620914425, 7.282396304205192, 7.868720572622599, 8.362558465796883, 8.932799826554965, 9.192904953214458, 9.767741663254332, 10.12080120237044, 10.60345412111397, 11.06608610586963, 11.50082175307439, 12.06430728593004, 12.52595148428033, 13.05728695053611

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