Properties

Label 2-91e2-1.1-c1-0-385
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-s − 1.41·3-s − 4-s + 2.68·5-s − 1.41·6-s − 3·8-s − 0.999·9-s + 2.68·10-s + 5.79·11-s + 1.41·12-s − 3.79·15-s − 16-s − 5.51·17-s − 0.999·18-s + 2.82·19-s − 2.68·20-s + 5.79·22-s + 1.79·23-s + 4.24·24-s + 2.20·25-s + 5.65·27-s − 8.79·29-s − 3.79·30-s + 1.41·31-s + 5·32-s − 8.19·33-s − 5.51·34-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.816·3-s − 0.5·4-s + 1.20·5-s − 0.577·6-s − 1.06·8-s − 0.333·9-s + 0.848·10-s + 1.74·11-s + 0.408·12-s − 0.980·15-s − 0.250·16-s − 1.33·17-s − 0.235·18-s + 0.648·19-s − 0.600·20-s + 1.23·22-s + 0.374·23-s + 0.866·24-s + 0.440·25-s + 1.08·27-s − 1.63·29-s − 0.693·30-s + 0.254·31-s + 0.883·32-s − 1.42·33-s − 0.945·34-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: \(1\)
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 \( 1 - T + 2T^{2} \)
3 \( 1 + 1.41T + 3T^{2} \)
5 \( 1 - 2.68T + 5T^{2} \)
11 \( 1 - 5.79T + 11T^{2} \)
17 \( 1 + 5.51T + 17T^{2} \)
19 \( 1 - 2.82T + 19T^{2} \)
23 \( 1 - 1.79T + 23T^{2} \)
29 \( 1 + 8.79T + 29T^{2} \)
31 \( 1 - 1.41T + 31T^{2} \)
37 \( 1 + 6.79T + 37T^{2} \)
41 \( 1 + 9.75T + 41T^{2} \)
43 \( 1 + 1.79T + 43T^{2} \)
47 \( 1 + 2.82T + 47T^{2} \)
53 \( 1 - 6.59T + 53T^{2} \)
59 \( 1 + 1.12T + 59T^{2} \)
61 \( 1 + 1.55T + 61T^{2} \)
67 \( 1 - 5.79T + 67T^{2} \)
71 \( 1 - 6T + 71T^{2} \)
73 \( 1 - 5.80T + 73T^{2} \)
79 \( 1 + 11.7T + 79T^{2} \)
83 \( 1 - 9.89T + 83T^{2} \)
89 \( 1 + 12.1T + 89T^{2} \)
97 \( 1 - 4.24T + 97T^{2} \)
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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

−6.93237896553180909720584645805, −6.59330730637756045535355660873, −5.93480318090382271415613534820, −5.38387949042911495030573417009, −4.86858001278640850079162306416, −3.97572472596881026942848007382, −3.32558283982703256973298074961, −2.19595322878054087625606661067, −1.28233310016117794014915772548, 0, 1.28233310016117794014915772548, 2.19595322878054087625606661067, 3.32558283982703256973298074961, 3.97572472596881026942848007382, 4.86858001278640850079162306416, 5.38387949042911495030573417009, 5.93480318090382271415613534820, 6.59330730637756045535355660873, 6.93237896553180909720584645805

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