Properties

Label 2-8015-1.1-c1-0-94
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.73·2-s − 2.59·3-s + 0.998·4-s − 5-s + 4.49·6-s − 7-s + 1.73·8-s + 3.73·9-s + 1.73·10-s − 5.80·11-s − 2.58·12-s − 4.81·13-s + 1.73·14-s + 2.59·15-s − 4.99·16-s − 6.09·17-s − 6.46·18-s − 1.57·19-s − 0.998·20-s + 2.59·21-s + 10.0·22-s + 3.03·23-s − 4.50·24-s + 25-s + 8.33·26-s − 1.89·27-s − 0.998·28-s + ⋯
L(s)  = 1  − 1.22·2-s − 1.49·3-s + 0.499·4-s − 0.447·5-s + 1.83·6-s − 0.377·7-s + 0.613·8-s + 1.24·9-s + 0.547·10-s − 1.74·11-s − 0.747·12-s − 1.33·13-s + 0.462·14-s + 0.669·15-s − 1.24·16-s − 1.47·17-s − 1.52·18-s − 0.362·19-s − 0.223·20-s + 0.566·21-s + 2.14·22-s + 0.631·23-s − 0.918·24-s + 0.200·25-s + 1.63·26-s − 0.365·27-s − 0.188·28-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: \(1\)
Selberg data: \((2,\ 8015,\ (\ :1/2),\ -1)\)

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)$
bad5 \( 1 + T \)
7 \( 1 + T \)
229 \( 1 + T \)
good2 \( 1 + 1.73T + 2T^{2} \)
3 \( 1 + 2.59T + 3T^{2} \)
11 \( 1 + 5.80T + 11T^{2} \)
13 \( 1 + 4.81T + 13T^{2} \)
17 \( 1 + 6.09T + 17T^{2} \)
19 \( 1 + 1.57T + 19T^{2} \)
23 \( 1 - 3.03T + 23T^{2} \)
29 \( 1 + 1.20T + 29T^{2} \)
31 \( 1 + 6.38T + 31T^{2} \)
37 \( 1 - 6.93T + 37T^{2} \)
41 \( 1 - 0.935T + 41T^{2} \)
43 \( 1 + 2.81T + 43T^{2} \)
47 \( 1 + 7.57T + 47T^{2} \)
53 \( 1 + 1.39T + 53T^{2} \)
59 \( 1 - 12.5T + 59T^{2} \)
61 \( 1 - 2.64T + 61T^{2} \)
67 \( 1 - 0.00740T + 67T^{2} \)
71 \( 1 + 1.56T + 71T^{2} \)
73 \( 1 + 16.2T + 73T^{2} \)
79 \( 1 - 9.63T + 79T^{2} \)
83 \( 1 + 4.78T + 83T^{2} \)
89 \( 1 - 9.59T + 89T^{2} \)
97 \( 1 + 14.1T + 97T^{2} \)
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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

−7.39075138223288752288414338679, −7.04739514645094886105725347410, −6.27201499984906517804737251310, −5.28740347465368884426289517919, −4.88755709036447697802827864940, −4.22208522147201605729179681539, −2.78370290743622181755195168497, −1.98459425214462918407996683821, −0.53530571312770429468022735048, 0, 0.53530571312770429468022735048, 1.98459425214462918407996683821, 2.78370290743622181755195168497, 4.22208522147201605729179681539, 4.88755709036447697802827864940, 5.28740347465368884426289517919, 6.27201499984906517804737251310, 7.04739514645094886105725347410, 7.39075138223288752288414338679

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