Properties

Label 2-91e2-1.1-c1-0-116
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  − 1.37·2-s + 2.88·3-s − 0.0982·4-s − 0.805·5-s − 3.97·6-s + 2.89·8-s + 5.30·9-s + 1.11·10-s − 5.27·11-s − 0.282·12-s − 2.32·15-s − 3.79·16-s − 0.560·17-s − 7.31·18-s − 5.84·19-s + 0.0791·20-s + 7.26·22-s − 1.60·23-s + 8.33·24-s − 4.35·25-s + 6.63·27-s + 2.28·29-s + 3.20·30-s − 3.47·31-s − 0.555·32-s − 15.1·33-s + 0.772·34-s + ⋯
L(s)  = 1  − 0.975·2-s + 1.66·3-s − 0.0491·4-s − 0.360·5-s − 1.62·6-s + 1.02·8-s + 1.76·9-s + 0.351·10-s − 1.58·11-s − 0.0816·12-s − 0.599·15-s − 0.948·16-s − 0.135·17-s − 1.72·18-s − 1.34·19-s + 0.0176·20-s + 1.54·22-s − 0.334·23-s + 1.70·24-s − 0.870·25-s + 1.27·27-s + 0.423·29-s + 0.584·30-s − 0.624·31-s − 0.0981·32-s − 2.64·33-s + 0.132·34-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

\(L(1)\) \(\approx\) \(1.310666002\)
\(L(\frac12)\) \(\approx\) \(1.310666002\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.37T + 2T^{2} \)
3 \( 1 - 2.88T + 3T^{2} \)
5 \( 1 + 0.805T + 5T^{2} \)
11 \( 1 + 5.27T + 11T^{2} \)
17 \( 1 + 0.560T + 17T^{2} \)
19 \( 1 + 5.84T + 19T^{2} \)
23 \( 1 + 1.60T + 23T^{2} \)
29 \( 1 - 2.28T + 29T^{2} \)
31 \( 1 + 3.47T + 31T^{2} \)
37 \( 1 + 1.24T + 37T^{2} \)
41 \( 1 + 0.927T + 41T^{2} \)
43 \( 1 - 4.44T + 43T^{2} \)
47 \( 1 - 3.84T + 47T^{2} \)
53 \( 1 - 5.45T + 53T^{2} \)
59 \( 1 - 10.9T + 59T^{2} \)
61 \( 1 - 7.30T + 61T^{2} \)
67 \( 1 - 7.34T + 67T^{2} \)
71 \( 1 + 9.31T + 71T^{2} \)
73 \( 1 - 5.00T + 73T^{2} \)
79 \( 1 - 11.3T + 79T^{2} \)
83 \( 1 - 5.81T + 83T^{2} \)
89 \( 1 + 5.00T + 89T^{2} \)
97 \( 1 - 10.6T + 97T^{2} \)
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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

−8.058871887771847967500202639934, −7.50346050184362950758165240785, −6.95858575474311712954810504002, −5.74104717599432555618204412331, −4.77567511511663291116436361521, −4.08977412108114971228063426524, −3.43606794926119468901468196237, −2.27590811178256396538740058483, −2.09510634200149993100592086143, −0.57628664883330566700845846464, 0.57628664883330566700845846464, 2.09510634200149993100592086143, 2.27590811178256396538740058483, 3.43606794926119468901468196237, 4.08977412108114971228063426524, 4.77567511511663291116436361521, 5.74104717599432555618204412331, 6.95858575474311712954810504002, 7.50346050184362950758165240785, 8.058871887771847967500202639934

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