Properties

Label 2-91e2-1.1-c1-0-463
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  + 2.61·2-s + 2.23·3-s + 4.85·4-s + 2.23·5-s + 5.85·6-s + 7.47·8-s + 2.00·9-s + 5.85·10-s + 3·11-s + 10.8·12-s + 5.00·15-s + 9.85·16-s − 1.47·17-s + 5.23·18-s + 3·19-s + 10.8·20-s + 7.85·22-s − 8.23·23-s + 16.7·24-s − 2.23·27-s + 4.47·29-s + 13.0·30-s + 5·31-s + 10.8·32-s + 6.70·33-s − 3.85·34-s + 9.70·36-s + ⋯
L(s)  = 1  + 1.85·2-s + 1.29·3-s + 2.42·4-s + 0.999·5-s + 2.38·6-s + 2.64·8-s + 0.666·9-s + 1.85·10-s + 0.904·11-s + 3.13·12-s + 1.29·15-s + 2.46·16-s − 0.357·17-s + 1.23·18-s + 0.688·19-s + 2.42·20-s + 1.67·22-s − 1.71·23-s + 3.41·24-s − 0.430·27-s + 0.830·29-s + 2.38·30-s + 0.898·31-s + 1.91·32-s + 1.16·33-s − 0.660·34-s + 1.61·36-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

\(L(1)\) \(\approx\) \(13.84337183\)
\(L(\frac12)\) \(\approx\) \(13.84337183\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.61T + 2T^{2} \)
3 \( 1 - 2.23T + 3T^{2} \)
5 \( 1 - 2.23T + 5T^{2} \)
11 \( 1 - 3T + 11T^{2} \)
17 \( 1 + 1.47T + 17T^{2} \)
19 \( 1 - 3T + 19T^{2} \)
23 \( 1 + 8.23T + 23T^{2} \)
29 \( 1 - 4.47T + 29T^{2} \)
31 \( 1 - 5T + 31T^{2} \)
37 \( 1 + 4.70T + 37T^{2} \)
41 \( 1 + 4.47T + 41T^{2} \)
43 \( 1 + 8T + 43T^{2} \)
47 \( 1 + 7.47T + 47T^{2} \)
53 \( 1 + 7.47T + 53T^{2} \)
59 \( 1 + 1.47T + 59T^{2} \)
61 \( 1 + 3T + 61T^{2} \)
67 \( 1 - 3T + 67T^{2} \)
71 \( 1 - 8.94T + 71T^{2} \)
73 \( 1 + 2.70T + 73T^{2} \)
79 \( 1 + 2.70T + 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 - 2.23T + 89T^{2} \)
97 \( 1 - 9.41T + 97T^{2} \)
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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

−7.73496349705862751558404028915, −6.69005494630249353699739735354, −6.41351457488829962799862626931, −5.65978229613637070655340511471, −4.89862679230168212830244837030, −4.17052913989804499796135433640, −3.44986878532506369747052712632, −2.94719526213056545435382393792, −2.01256227550936978261495537370, −1.66539717911064327220235980572, 1.66539717911064327220235980572, 2.01256227550936978261495537370, 2.94719526213056545435382393792, 3.44986878532506369747052712632, 4.17052913989804499796135433640, 4.89862679230168212830244837030, 5.65978229613637070655340511471, 6.41351457488829962799862626931, 6.69005494630249353699739735354, 7.73496349705862751558404028915

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