Properties

Label 2-8015-1.1-c1-0-350
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  + 1.82·2-s − 2.85·3-s + 1.32·4-s + 5-s − 5.21·6-s + 7-s − 1.23·8-s + 5.16·9-s + 1.82·10-s − 0.0862·11-s − 3.78·12-s − 0.762·13-s + 1.82·14-s − 2.85·15-s − 4.89·16-s − 0.690·17-s + 9.42·18-s + 0.149·19-s + 1.32·20-s − 2.85·21-s − 0.157·22-s − 0.700·23-s + 3.52·24-s + 25-s − 1.39·26-s − 6.20·27-s + 1.32·28-s + ⋯
L(s)  = 1  + 1.28·2-s − 1.65·3-s + 0.662·4-s + 0.447·5-s − 2.12·6-s + 0.377·7-s − 0.435·8-s + 1.72·9-s + 0.576·10-s − 0.0260·11-s − 1.09·12-s − 0.211·13-s + 0.487·14-s − 0.738·15-s − 1.22·16-s − 0.167·17-s + 2.22·18-s + 0.0343·19-s + 0.296·20-s − 0.623·21-s − 0.0335·22-s − 0.146·23-s + 0.718·24-s + 0.200·25-s − 0.272·26-s − 1.19·27-s + 0.250·28-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 \( 1 - 1.82T + 2T^{2} \)
3 \( 1 + 2.85T + 3T^{2} \)
11 \( 1 + 0.0862T + 11T^{2} \)
13 \( 1 + 0.762T + 13T^{2} \)
17 \( 1 + 0.690T + 17T^{2} \)
19 \( 1 - 0.149T + 19T^{2} \)
23 \( 1 + 0.700T + 23T^{2} \)
29 \( 1 - 0.886T + 29T^{2} \)
31 \( 1 - 5.07T + 31T^{2} \)
37 \( 1 + 6.12T + 37T^{2} \)
41 \( 1 - 1.80T + 41T^{2} \)
43 \( 1 - 11.7T + 43T^{2} \)
47 \( 1 + 1.88T + 47T^{2} \)
53 \( 1 + 7.84T + 53T^{2} \)
59 \( 1 + 8.01T + 59T^{2} \)
61 \( 1 + 0.455T + 61T^{2} \)
67 \( 1 + 8.65T + 67T^{2} \)
71 \( 1 + 13.8T + 71T^{2} \)
73 \( 1 - 7.74T + 73T^{2} \)
79 \( 1 + 3.32T + 79T^{2} \)
83 \( 1 - 14.0T + 83T^{2} \)
89 \( 1 + 8.87T + 89T^{2} \)
97 \( 1 - 3.04T + 97T^{2} \)
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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

−7.04845143078760868608327697713, −6.43178991301319939838987455016, −5.88589571166975740981974477107, −5.44040575888269343960501581750, −4.62462607351225883543814267217, −4.45239481583231202331360876829, −3.32019417359286756399686498036, −2.34141295111498945287706158741, −1.23349368043112522581354222494, 0, 1.23349368043112522581354222494, 2.34141295111498945287706158741, 3.32019417359286756399686498036, 4.45239481583231202331360876829, 4.62462607351225883543814267217, 5.44040575888269343960501581750, 5.88589571166975740981974477107, 6.43178991301319939838987455016, 7.04845143078760868608327697713

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