Properties

Label 2-91e2-1.1-c1-0-206
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·4-s − 3.60·5-s − 3·9-s + 4·16-s − 3.60·19-s + 7.21·20-s − 23-s + 7.99·25-s − 5·29-s + 10.8·31-s + 6·36-s − 7.21·41-s − 9·43-s + 10.8·45-s + 3.60·47-s + 11·53-s + 14.4·59-s − 8·64-s − 10.8·73-s + 7.21·76-s + 15·79-s − 14.4·80-s + 9·81-s + 18.0·83-s + 3.60·89-s + 2·92-s + 12.9·95-s + ⋯
L(s)  = 1  − 4-s − 1.61·5-s − 9-s + 16-s − 0.827·19-s + 1.61·20-s − 0.208·23-s + 1.59·25-s − 0.928·29-s + 1.94·31-s + 36-s − 1.12·41-s − 1.37·43-s + 1.61·45-s + 0.525·47-s + 1.51·53-s + 1.87·59-s − 64-s − 1.26·73-s + 0.827·76-s + 1.68·79-s − 1.61·80-s + 81-s + 1.97·83-s + 0.382·89-s + 0.208·92-s + 1.33·95-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: \(1\)
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 \( 1 + 2T^{2} \)
3 \( 1 + 3T^{2} \)
5 \( 1 + 3.60T + 5T^{2} \)
11 \( 1 + 11T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 + 3.60T + 19T^{2} \)
23 \( 1 + T + 23T^{2} \)
29 \( 1 + 5T + 29T^{2} \)
31 \( 1 - 10.8T + 31T^{2} \)
37 \( 1 + 37T^{2} \)
41 \( 1 + 7.21T + 41T^{2} \)
43 \( 1 + 9T + 43T^{2} \)
47 \( 1 - 3.60T + 47T^{2} \)
53 \( 1 - 11T + 53T^{2} \)
59 \( 1 - 14.4T + 59T^{2} \)
61 \( 1 + 61T^{2} \)
67 \( 1 + 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 10.8T + 73T^{2} \)
79 \( 1 - 15T + 79T^{2} \)
83 \( 1 - 18.0T + 83T^{2} \)
89 \( 1 - 3.60T + 89T^{2} \)
97 \( 1 - 18.0T + 97T^{2} \)
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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

−7.69913170685552903296299653110, −6.84881867125274396872993919539, −6.05860702388404523739010663704, −5.14653537581162888608739552742, −4.63096275524713101713774358868, −3.78647866064763939135712329726, −3.44662570850893324070288468305, −2.37350425892920205370682200506, −0.800506457038119216787322834861, 0, 0.800506457038119216787322834861, 2.37350425892920205370682200506, 3.44662570850893324070288468305, 3.78647866064763939135712329726, 4.63096275524713101713774358868, 5.14653537581162888608739552742, 6.05860702388404523739010663704, 6.84881867125274396872993919539, 7.69913170685552903296299653110

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