Properties

Label 2-296208-1.1-c1-0-105
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  + 5-s − 2·7-s + 5·13-s − 17-s − 19-s − 7·23-s − 4·25-s + 6·29-s − 4·31-s − 2·35-s + 10·37-s + 9·41-s + 43-s − 12·47-s − 3·49-s + 12·53-s + 6·59-s − 2·61-s + 5·65-s − 4·67-s − 8·71-s − 6·79-s − 4·83-s − 85-s − 2·89-s − 10·91-s − 95-s + ⋯
L(s)  = 1  + 0.447·5-s − 0.755·7-s + 1.38·13-s − 0.242·17-s − 0.229·19-s − 1.45·23-s − 4/5·25-s + 1.11·29-s − 0.718·31-s − 0.338·35-s + 1.64·37-s + 1.40·41-s + 0.152·43-s − 1.75·47-s − 3/7·49-s + 1.64·53-s + 0.781·59-s − 0.256·61-s + 0.620·65-s − 0.488·67-s − 0.949·71-s − 0.675·79-s − 0.439·83-s − 0.108·85-s − 0.211·89-s − 1.04·91-s − 0.102·95-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 \( 1 - T + p T^{2} \) 1.5.ab
7 \( 1 + 2 T + p T^{2} \) 1.7.c
13 \( 1 - 5 T + p T^{2} \) 1.13.af
19 \( 1 + T + p T^{2} \) 1.19.b
23 \( 1 + 7 T + p T^{2} \) 1.23.h
29 \( 1 - 6 T + p T^{2} \) 1.29.ag
31 \( 1 + 4 T + p T^{2} \) 1.31.e
37 \( 1 - 10 T + p T^{2} \) 1.37.ak
41 \( 1 - 9 T + p T^{2} \) 1.41.aj
43 \( 1 - T + p T^{2} \) 1.43.ab
47 \( 1 + 12 T + p T^{2} \) 1.47.m
53 \( 1 - 12 T + p T^{2} \) 1.53.am
59 \( 1 - 6 T + p T^{2} \) 1.59.ag
61 \( 1 + 2 T + p T^{2} \) 1.61.c
67 \( 1 + 4 T + p T^{2} \) 1.67.e
71 \( 1 + 8 T + p T^{2} \) 1.71.i
73 \( 1 + p T^{2} \) 1.73.a
79 \( 1 + 6 T + p T^{2} \) 1.79.g
83 \( 1 + 4 T + p T^{2} \) 1.83.e
89 \( 1 + 2 T + p T^{2} \) 1.89.c
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

−12.87074657367544, −12.79320284187444, −11.88014761323964, −11.51553387671655, −11.25604014739953, −10.39833731667759, −10.26833819971692, −9.737946148566706, −9.308027755419579, −8.738941245389600, −8.422076384074418, −7.774860236898491, −7.447137323186974, −6.587008564764510, −6.340238120527611, −5.866081852379237, −5.668374158291025, −4.705480490291384, −4.260091844790690, −3.764514427833090, −3.289999734455752, −2.590213763855256, −2.117081823713344, −1.445062664050849, −0.7864376196382707, 0, 0.7864376196382707, 1.445062664050849, 2.117081823713344, 2.590213763855256, 3.289999734455752, 3.764514427833090, 4.260091844790690, 4.705480490291384, 5.668374158291025, 5.866081852379237, 6.340238120527611, 6.587008564764510, 7.447137323186974, 7.774860236898491, 8.422076384074418, 8.738941245389600, 9.308027755419579, 9.737946148566706, 10.26833819971692, 10.39833731667759, 11.25604014739953, 11.51553387671655, 11.88014761323964, 12.79320284187444, 12.87074657367544

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