Properties

Label 2-91e2-1.1-c1-0-75
Degree $2$
Conductor $8281$
Sign $1$
Analytic cond. $66.1241$
Root an. cond. $8.13167$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.499·2-s + 0.849·3-s − 1.75·4-s + 1.04·5-s − 0.424·6-s + 1.87·8-s − 2.27·9-s − 0.521·10-s − 3.96·11-s − 1.48·12-s + 0.885·15-s + 2.56·16-s − 0.142·17-s + 1.13·18-s − 5.50·19-s − 1.82·20-s + 1.98·22-s + 4.39·23-s + 1.59·24-s − 3.91·25-s − 4.48·27-s − 8.39·29-s − 0.442·30-s + 2.84·31-s − 5.03·32-s − 3.37·33-s + 0.0710·34-s + ⋯
L(s)  = 1  − 0.353·2-s + 0.490·3-s − 0.875·4-s + 0.466·5-s − 0.173·6-s + 0.662·8-s − 0.759·9-s − 0.164·10-s − 1.19·11-s − 0.429·12-s + 0.228·15-s + 0.640·16-s − 0.0344·17-s + 0.268·18-s − 1.26·19-s − 0.407·20-s + 0.422·22-s + 0.915·23-s + 0.325·24-s − 0.782·25-s − 0.863·27-s − 1.55·29-s − 0.0808·30-s + 0.511·31-s − 0.889·32-s − 0.586·33-s + 0.0121·34-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: no
Arithmetic: yes
Character: $\chi_{8281} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 8281,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8995412584\)
\(L(\frac12)\) \(\approx\) \(0.8995412584\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.499T + 2T^{2} \)
3 \( 1 - 0.849T + 3T^{2} \)
5 \( 1 - 1.04T + 5T^{2} \)
11 \( 1 + 3.96T + 11T^{2} \)
17 \( 1 + 0.142T + 17T^{2} \)
19 \( 1 + 5.50T + 19T^{2} \)
23 \( 1 - 4.39T + 23T^{2} \)
29 \( 1 + 8.39T + 29T^{2} \)
31 \( 1 - 2.84T + 31T^{2} \)
37 \( 1 + 0.843T + 37T^{2} \)
41 \( 1 - 12.0T + 41T^{2} \)
43 \( 1 - 4.82T + 43T^{2} \)
47 \( 1 + 4.55T + 47T^{2} \)
53 \( 1 + 0.279T + 53T^{2} \)
59 \( 1 + 10.7T + 59T^{2} \)
61 \( 1 + 5.86T + 61T^{2} \)
67 \( 1 - 5.14T + 67T^{2} \)
71 \( 1 - 3.69T + 71T^{2} \)
73 \( 1 + 6.61T + 73T^{2} \)
79 \( 1 - 11.9T + 79T^{2} \)
83 \( 1 - 2.87T + 83T^{2} \)
89 \( 1 - 1.74T + 89T^{2} \)
97 \( 1 + 2.70T + 97T^{2} \)
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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

−7.76993303893958568679939620218, −7.59439169368486928310270767555, −6.27600239875506394640411466372, −5.70965473477323099014315989814, −5.03874795594412049151232098529, −4.29166227657177061196240614483, −3.44273788055907398984862097680, −2.57830176233681423811229416776, −1.86021985350655270774144879825, −0.46398386607818375891804417352, 0.46398386607818375891804417352, 1.86021985350655270774144879825, 2.57830176233681423811229416776, 3.44273788055907398984862097680, 4.29166227657177061196240614483, 5.03874795594412049151232098529, 5.70965473477323099014315989814, 6.27600239875506394640411466372, 7.59439169368486928310270767555, 7.76993303893958568679939620218

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