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Properties

Label 2-81232-1.1-c1-0-3
Degree $2$
Conductor $81232$
Sign $1$
Analytic cond. $648.640$
Root an. cond. $25.4684$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $2$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 2·9-s − 4·13-s − 2·17-s + 19-s − 6·23-s − 5·25-s − 5·27-s − 8·29-s + 4·31-s − 2·37-s − 4·39-s − 10·41-s + 8·43-s − 9·47-s − 7·49-s − 2·51-s − 53-s + 57-s + 9·59-s − 10·61-s + 4·67-s − 6·69-s − 12·71-s − 16·73-s − 5·75-s + 13·79-s + ⋯
L(s)  = 1  + 0.577·3-s − 2/3·9-s − 1.10·13-s − 0.485·17-s + 0.229·19-s − 1.25·23-s − 25-s − 0.962·27-s − 1.48·29-s + 0.718·31-s − 0.328·37-s − 0.640·39-s − 1.56·41-s + 1.21·43-s − 1.31·47-s − 49-s − 0.280·51-s − 0.137·53-s + 0.132·57-s + 1.17·59-s − 1.28·61-s + 0.488·67-s − 0.722·69-s − 1.42·71-s − 1.87·73-s − 0.577·75-s + 1.46·79-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 81232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81232 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(81232\)    =    \(2^{4} \cdot 5077\)
Sign: $1$
Analytic conductor: \(648.640\)
Root analytic conductor: \(25.4684\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((2,\ 81232,\ (\ :1/2),\ 1)\)

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)$
bad2 \( 1 \)
5077 \( 1 - T \)
good3 \( 1 - T + p T^{2} \)
5 \( 1 + p T^{2} \)
7 \( 1 + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + 9 T + p T^{2} \)
53 \( 1 + T + p T^{2} \)
59 \( 1 - 9 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 16 T + p T^{2} \)
79 \( 1 - 13 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 13 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−14.45705826812149, −14.05894587876080, −13.43319125119354, −13.22209871341734, −12.42671137644284, −11.97341296411442, −11.51679369240338, −11.15891625578781, −10.29134061589708, −9.943620750300799, −9.422274356790527, −9.020764085893267, −8.263177573073402, −7.998292552795771, −7.466783169370006, −6.857794797436363, −6.249512056525379, −5.583772412766411, −5.266053119738317, −4.361475247873599, −3.998070806230749, −3.185169047069432, −2.748578902649732, −2.001780159690983, −1.589333567092025, 0, 0, 1.589333567092025, 2.001780159690983, 2.748578902649732, 3.185169047069432, 3.998070806230749, 4.361475247873599, 5.266053119738317, 5.583772412766411, 6.249512056525379, 6.857794797436363, 7.466783169370006, 7.998292552795771, 8.263177573073402, 9.020764085893267, 9.422274356790527, 9.943620750300799, 10.29134061589708, 11.15891625578781, 11.51679369240338, 11.97341296411442, 12.42671137644284, 13.22209871341734, 13.43319125119354, 14.05894587876080, 14.45705826812149

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