Properties

Label 2-8037-1.1-c1-0-311
Degree $2$
Conductor $8037$
Sign $-1$
Analytic cond. $64.1757$
Root an. cond. $8.01097$
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  + 1.38·2-s − 0.0816·4-s + 2.56·5-s + 0.393·7-s − 2.88·8-s + 3.55·10-s − 0.0100·11-s − 1.56·13-s + 0.544·14-s − 3.82·16-s − 0.120·17-s − 19-s − 0.209·20-s − 0.0139·22-s + 3.27·23-s + 1.59·25-s − 2.17·26-s − 0.0321·28-s − 9.33·29-s − 4.52·31-s + 0.461·32-s − 0.167·34-s + 1.00·35-s − 2.88·37-s − 1.38·38-s − 7.40·40-s + 7.99·41-s + ⋯
L(s)  = 1  + 0.979·2-s − 0.0408·4-s + 1.14·5-s + 0.148·7-s − 1.01·8-s + 1.12·10-s − 0.00303·11-s − 0.435·13-s + 0.145·14-s − 0.957·16-s − 0.0292·17-s − 0.229·19-s − 0.0468·20-s − 0.00297·22-s + 0.682·23-s + 0.318·25-s − 0.426·26-s − 0.00606·28-s − 1.73·29-s − 0.813·31-s + 0.0816·32-s − 0.0286·34-s + 0.170·35-s − 0.473·37-s − 0.224·38-s − 1.17·40-s + 1.24·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8037 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8037 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8037\)    =    \(3^{2} \cdot 19 \cdot 47\)
Sign: $-1$
Analytic conductor: \(64.1757\)
Root analytic conductor: \(8.01097\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 8037,\ (\ :1/2),\ -1)\)

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)$
bad3 \( 1 \)
19 \( 1 + T \)
47 \( 1 - T \)
good2 \( 1 - 1.38T + 2T^{2} \)
5 \( 1 - 2.56T + 5T^{2} \)
7 \( 1 - 0.393T + 7T^{2} \)
11 \( 1 + 0.0100T + 11T^{2} \)
13 \( 1 + 1.56T + 13T^{2} \)
17 \( 1 + 0.120T + 17T^{2} \)
23 \( 1 - 3.27T + 23T^{2} \)
29 \( 1 + 9.33T + 29T^{2} \)
31 \( 1 + 4.52T + 31T^{2} \)
37 \( 1 + 2.88T + 37T^{2} \)
41 \( 1 - 7.99T + 41T^{2} \)
43 \( 1 + 11.2T + 43T^{2} \)
53 \( 1 + 1.80T + 53T^{2} \)
59 \( 1 - 8.28T + 59T^{2} \)
61 \( 1 + 7.98T + 61T^{2} \)
67 \( 1 - 16.2T + 67T^{2} \)
71 \( 1 + 14.5T + 71T^{2} \)
73 \( 1 + 0.336T + 73T^{2} \)
79 \( 1 + 16.0T + 79T^{2} \)
83 \( 1 - 14.2T + 83T^{2} \)
89 \( 1 + 9.82T + 89T^{2} \)
97 \( 1 + 17.1T + 97T^{2} \)
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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

−7.25259195461348344127364402696, −6.61012002161218123560877853085, −5.79997732866965055932939467725, −5.42963471871886082901226272766, −4.79424216650040477333327080655, −3.96832766044526519719050830146, −3.20503803967559737853729619829, −2.35062795429821596571641466441, −1.56511543658422838313663948349, 0, 1.56511543658422838313663948349, 2.35062795429821596571641466441, 3.20503803967559737853729619829, 3.96832766044526519719050830146, 4.79424216650040477333327080655, 5.42963471871886082901226272766, 5.79997732866965055932939467725, 6.61012002161218123560877853085, 7.25259195461348344127364402696

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