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Properties

Label 2-91e2-1.1-c1-0-316
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 − 3·3-s − 4-s + 3·5-s − 3·6-s − 3·8-s + 6·9-s + 3·10-s − 3·11-s + 3·12-s − 9·15-s − 16-s − 2·17-s + 6·18-s − 19-s − 3·20-s − 3·22-s + 9·24-s + 4·25-s − 9·27-s + 7·29-s − 9·30-s + 3·31-s + 5·32-s + 9·33-s − 2·34-s − 6·36-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.73·3-s − 1/2·4-s + 1.34·5-s − 1.22·6-s − 1.06·8-s + 2·9-s + 0.948·10-s − 0.904·11-s + 0.866·12-s − 2.32·15-s − 1/4·16-s − 0.485·17-s + 1.41·18-s − 0.229·19-s − 0.670·20-s − 0.639·22-s + 1.83·24-s + 4/5·25-s − 1.73·27-s + 1.29·29-s − 1.64·30-s + 0.538·31-s + 0.883·32-s + 1.56·33-s − 0.342·34-s − 36-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 \( 1 + p T + p T^{2} \)
5 \( 1 - 3 T + p T^{2} \)
11 \( 1 + 3 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 7 T + p T^{2} \)
31 \( 1 - 3 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 3 T + p T^{2} \)
43 \( 1 + 7 T + p T^{2} \)
47 \( 1 - T + p T^{2} \)
53 \( 1 - 3 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 13 T + p T^{2} \)
67 \( 1 + 3 T + p T^{2} \)
71 \( 1 - 13 T + p T^{2} \)
73 \( 1 + 13 T + p T^{2} \)
79 \( 1 + 3 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 5 T + p T^{2} \)
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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

−7.05362532194932532109578552471, −6.30367620644786691602532205020, −5.99169125898668912910185184504, −5.41357232391496343740671029659, −4.74921498859754640487471102963, −4.45138766055725594219682573053, −3.14255396590887286261559539361, −2.21363063969416839678289895564, −1.06882102719592073190368177529, 0, 1.06882102719592073190368177529, 2.21363063969416839678289895564, 3.14255396590887286261559539361, 4.45138766055725594219682573053, 4.74921498859754640487471102963, 5.41357232391496343740671029659, 5.99169125898668912910185184504, 6.30367620644786691602532205020, 7.05362532194932532109578552471

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