Properties

Label 2-8015-1.1-c1-0-50
Degree $2$
Conductor $8015$
Sign $1$
Analytic cond. $64.0000$
Root an. cond. $8.00000$
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  + 2.00·2-s − 2.97·3-s + 2.02·4-s − 5-s − 5.96·6-s − 7-s + 0.0406·8-s + 5.84·9-s − 2.00·10-s − 4.21·11-s − 6.00·12-s + 0.537·13-s − 2.00·14-s + 2.97·15-s − 3.95·16-s + 5.04·17-s + 11.7·18-s − 1.80·19-s − 2.02·20-s + 2.97·21-s − 8.45·22-s + 0.0595·23-s − 0.121·24-s + 25-s + 1.07·26-s − 8.47·27-s − 2.02·28-s + ⋯
L(s)  = 1  + 1.41·2-s − 1.71·3-s + 1.01·4-s − 0.447·5-s − 2.43·6-s − 0.377·7-s + 0.0143·8-s + 1.94·9-s − 0.634·10-s − 1.27·11-s − 1.73·12-s + 0.149·13-s − 0.535·14-s + 0.768·15-s − 0.989·16-s + 1.22·17-s + 2.76·18-s − 0.414·19-s − 0.451·20-s + 0.649·21-s − 1.80·22-s + 0.0124·23-s − 0.0247·24-s + 0.200·25-s + 0.211·26-s − 1.63·27-s − 0.381·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8015\)    =    \(5 \cdot 7 \cdot 229\)
Sign: $1$
Analytic conductor: \(64.0000\)
Root analytic conductor: \(8.00000\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 8015,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9472605229\)
\(L(\frac12)\) \(\approx\) \(0.9472605229\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + T \)
7 \( 1 + T \)
229 \( 1 - T \)
good2 \( 1 - 2.00T + 2T^{2} \)
3 \( 1 + 2.97T + 3T^{2} \)
11 \( 1 + 4.21T + 11T^{2} \)
13 \( 1 - 0.537T + 13T^{2} \)
17 \( 1 - 5.04T + 17T^{2} \)
19 \( 1 + 1.80T + 19T^{2} \)
23 \( 1 - 0.0595T + 23T^{2} \)
29 \( 1 - 3.07T + 29T^{2} \)
31 \( 1 + 3.60T + 31T^{2} \)
37 \( 1 + 10.8T + 37T^{2} \)
41 \( 1 + 4.34T + 41T^{2} \)
43 \( 1 + 7.91T + 43T^{2} \)
47 \( 1 + 2.71T + 47T^{2} \)
53 \( 1 + 6.21T + 53T^{2} \)
59 \( 1 + 3.72T + 59T^{2} \)
61 \( 1 + 2.31T + 61T^{2} \)
67 \( 1 - 1.76T + 67T^{2} \)
71 \( 1 + 6.64T + 71T^{2} \)
73 \( 1 - 3.46T + 73T^{2} \)
79 \( 1 - 16.1T + 79T^{2} \)
83 \( 1 - 9.34T + 83T^{2} \)
89 \( 1 - 12.1T + 89T^{2} \)
97 \( 1 - 4.40T + 97T^{2} \)
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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

−7.49056716589637852701688246962, −6.70875239498906454815698736308, −6.28821114202323138488021430906, −5.46389915626220202690253016606, −5.14519280466600634785049404786, −4.62752249596055567130429683525, −3.62621838628114094355695039785, −3.12881798675053798162412780698, −1.82732682126059409024221054596, −0.40869257476581157000498185597, 0.40869257476581157000498185597, 1.82732682126059409024221054596, 3.12881798675053798162412780698, 3.62621838628114094355695039785, 4.62752249596055567130429683525, 5.14519280466600634785049404786, 5.46389915626220202690253016606, 6.28821114202323138488021430906, 6.70875239498906454815698736308, 7.49056716589637852701688246962

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