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Properties

Label 2-91e2-1.1-c1-0-397
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  − 2·2-s + 2·3-s + 2·4-s + 5-s − 4·6-s + 9-s − 2·10-s + 2·11-s + 4·12-s + 2·15-s − 4·16-s − 6·17-s − 2·18-s − 3·19-s + 2·20-s − 4·22-s + 3·23-s − 4·25-s − 4·27-s + 3·29-s − 4·30-s − 3·31-s + 8·32-s + 4·33-s + 12·34-s + 2·36-s − 6·37-s + ⋯
L(s)  = 1  − 1.41·2-s + 1.15·3-s + 4-s + 0.447·5-s − 1.63·6-s + 1/3·9-s − 0.632·10-s + 0.603·11-s + 1.15·12-s + 0.516·15-s − 16-s − 1.45·17-s − 0.471·18-s − 0.688·19-s + 0.447·20-s − 0.852·22-s + 0.625·23-s − 4/5·25-s − 0.769·27-s + 0.557·29-s − 0.730·30-s − 0.538·31-s + 1.41·32-s + 0.696·33-s + 2.05·34-s + 1/3·36-s − 0.986·37-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 \( 1 + p T + p T^{2} \)
3 \( 1 - 2 T + p T^{2} \)
5 \( 1 - T + p T^{2} \)
11 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 3 T + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 + 3 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 + T + p T^{2} \)
47 \( 1 - 11 T + p T^{2} \)
53 \( 1 + 9 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 14 T + p T^{2} \)
73 \( 1 + 9 T + p T^{2} \)
79 \( 1 + 9 T + p T^{2} \)
83 \( 1 - 11 T + p T^{2} \)
89 \( 1 + 5 T + p T^{2} \)
97 \( 1 + 9 T + p T^{2} \)
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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

−7.79446171704116925379246403228, −6.94050611254691063243903363127, −6.52029230632783603195129322117, −5.51164982715112344036735995949, −4.40536271624864945428650143987, −3.81313647633281493987954838326, −2.60593773950640727074396089643, −2.16557865487257437883082983963, −1.31884250620145067984233447932, 0, 1.31884250620145067984233447932, 2.16557865487257437883082983963, 2.60593773950640727074396089643, 3.81313647633281493987954838326, 4.40536271624864945428650143987, 5.51164982715112344036735995949, 6.52029230632783603195129322117, 6.94050611254691063243903363127, 7.79446171704116925379246403228

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