Properties

Label 2-91e2-1.1-c1-0-331
Degree $2$
Conductor $8281$
Sign $-1$
Analytic cond. $66.1241$
Root an. cond. $8.13167$
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-s − 4-s + 3·8-s − 3·9-s + 3·11-s − 16-s + 7·17-s + 3·18-s + 7·19-s − 3·22-s − 6·23-s − 5·25-s − 5·29-s − 5·32-s − 7·34-s + 3·36-s − 8·37-s − 7·38-s + 2·43-s − 3·44-s + 6·46-s − 7·47-s + 5·50-s − 3·53-s + 5·58-s + 7·59-s − 7·61-s + ⋯
L(s)  = 1  − 0.707·2-s − 1/2·4-s + 1.06·8-s − 9-s + 0.904·11-s − 1/4·16-s + 1.69·17-s + 0.707·18-s + 1.60·19-s − 0.639·22-s − 1.25·23-s − 25-s − 0.928·29-s − 0.883·32-s − 1.20·34-s + 1/2·36-s − 1.31·37-s − 1.13·38-s + 0.304·43-s − 0.452·44-s + 0.884·46-s − 1.02·47-s + 0.707·50-s − 0.412·53-s + 0.656·58-s + 0.911·59-s − 0.896·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8281 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8281 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8281\)    =    \(7^{2} \cdot 13^{2}\)
Sign: $-1$
Analytic conductor: \(66.1241\)
Root analytic conductor: \(8.13167\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 8281,\ (\ :1/2),\ -1)\)

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)$
bad7 \( 1 \)
13 \( 1 \)
good2 \( 1 + T + p T^{2} \)
3 \( 1 + p T^{2} \)
5 \( 1 + p T^{2} \)
11 \( 1 - 3 T + p T^{2} \)
17 \( 1 - 7 T + p T^{2} \)
19 \( 1 - 7 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 5 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 + 7 T + p T^{2} \)
53 \( 1 + 3 T + p T^{2} \)
59 \( 1 - 7 T + p T^{2} \)
61 \( 1 + 7 T + p T^{2} \)
67 \( 1 - 3 T + p T^{2} \)
71 \( 1 - 5 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 + 6 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + p T^{2} \)
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

−7.72020344451817366780446081149, −7.02083974777780968223389730777, −5.81236310630135789832455931525, −5.62182358465587193429334467212, −4.69617021715968351952529324378, −3.63962172778598183251155908761, −3.34118417062116928652202893072, −1.91522785010362047382711078400, −1.11606055452145369890840422004, 0, 1.11606055452145369890840422004, 1.91522785010362047382711078400, 3.34118417062116928652202893072, 3.63962172778598183251155908761, 4.69617021715968351952529324378, 5.62182358465587193429334467212, 5.81236310630135789832455931525, 7.02083974777780968223389730777, 7.72020344451817366780446081149

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