Properties

Label 2-8015-1.1-c1-0-451
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  + 1.93·2-s + 1.65·3-s + 1.76·4-s + 5-s + 3.21·6-s − 7-s − 0.465·8-s − 0.247·9-s + 1.93·10-s + 4.68·11-s + 2.91·12-s − 4.69·13-s − 1.93·14-s + 1.65·15-s − 4.42·16-s − 2.83·17-s − 0.480·18-s − 5.47·19-s + 1.76·20-s − 1.65·21-s + 9.08·22-s − 7.64·23-s − 0.771·24-s + 25-s − 9.10·26-s − 5.38·27-s − 1.76·28-s + ⋯
L(s)  = 1  + 1.37·2-s + 0.957·3-s + 0.880·4-s + 0.447·5-s + 1.31·6-s − 0.377·7-s − 0.164·8-s − 0.0825·9-s + 0.613·10-s + 1.41·11-s + 0.842·12-s − 1.30·13-s − 0.518·14-s + 0.428·15-s − 1.10·16-s − 0.686·17-s − 0.113·18-s − 1.25·19-s + 0.393·20-s − 0.362·21-s + 1.93·22-s − 1.59·23-s − 0.157·24-s + 0.200·25-s − 1.78·26-s − 1.03·27-s − 0.332·28-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 \( 1 - 1.93T + 2T^{2} \)
3 \( 1 - 1.65T + 3T^{2} \)
11 \( 1 - 4.68T + 11T^{2} \)
13 \( 1 + 4.69T + 13T^{2} \)
17 \( 1 + 2.83T + 17T^{2} \)
19 \( 1 + 5.47T + 19T^{2} \)
23 \( 1 + 7.64T + 23T^{2} \)
29 \( 1 + 0.778T + 29T^{2} \)
31 \( 1 + 5.95T + 31T^{2} \)
37 \( 1 + 0.530T + 37T^{2} \)
41 \( 1 + 2.86T + 41T^{2} \)
43 \( 1 - 9.21T + 43T^{2} \)
47 \( 1 - 0.554T + 47T^{2} \)
53 \( 1 - 4.18T + 53T^{2} \)
59 \( 1 - 6.08T + 59T^{2} \)
61 \( 1 + 2.57T + 61T^{2} \)
67 \( 1 - 11.4T + 67T^{2} \)
71 \( 1 + 1.01T + 71T^{2} \)
73 \( 1 + 10.1T + 73T^{2} \)
79 \( 1 - 3.22T + 79T^{2} \)
83 \( 1 + 11.4T + 83T^{2} \)
89 \( 1 - 5.65T + 89T^{2} \)
97 \( 1 + 3.42T + 97T^{2} \)
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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

−7.21092280258796612024036288809, −6.66846256442355028130969671228, −5.97553006065512308100190472826, −5.43235155847158846528767761314, −4.26386475298948092831357971919, −4.09744283500829143002713825321, −3.23603867306454263077227909694, −2.33050746689012284326443104947, −2.00396324901629653069264247870, 0, 2.00396324901629653069264247870, 2.33050746689012284326443104947, 3.23603867306454263077227909694, 4.09744283500829143002713825321, 4.26386475298948092831357971919, 5.43235155847158846528767761314, 5.97553006065512308100190472826, 6.66846256442355028130969671228, 7.21092280258796612024036288809

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