Properties

Label 2-8007-1.1-c1-0-213
Degree $2$
Conductor $8007$
Sign $-1$
Analytic cond. $63.9362$
Root an. cond. $7.99601$
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  − 0.744·2-s − 3-s − 1.44·4-s + 1.71·5-s + 0.744·6-s − 3.69·7-s + 2.56·8-s + 9-s − 1.27·10-s − 0.320·11-s + 1.44·12-s − 1.38·13-s + 2.74·14-s − 1.71·15-s + 0.983·16-s + 17-s − 0.744·18-s − 1.35·19-s − 2.47·20-s + 3.69·21-s + 0.238·22-s + 3.98·23-s − 2.56·24-s − 2.05·25-s + 1.02·26-s − 27-s + 5.33·28-s + ⋯
L(s)  = 1  − 0.526·2-s − 0.577·3-s − 0.723·4-s + 0.766·5-s + 0.303·6-s − 1.39·7-s + 0.906·8-s + 0.333·9-s − 0.403·10-s − 0.0967·11-s + 0.417·12-s − 0.383·13-s + 0.734·14-s − 0.442·15-s + 0.245·16-s + 0.242·17-s − 0.175·18-s − 0.311·19-s − 0.554·20-s + 0.805·21-s + 0.0508·22-s + 0.831·23-s − 0.523·24-s − 0.411·25-s + 0.201·26-s − 0.192·27-s + 1.00·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8007\)    =    \(3 \cdot 17 \cdot 157\)
Sign: $-1$
Analytic conductor: \(63.9362\)
Root analytic conductor: \(7.99601\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 8007,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
17 \( 1 - T \)
157 \( 1 - T \)
good2 \( 1 + 0.744T + 2T^{2} \)
5 \( 1 - 1.71T + 5T^{2} \)
7 \( 1 + 3.69T + 7T^{2} \)
11 \( 1 + 0.320T + 11T^{2} \)
13 \( 1 + 1.38T + 13T^{2} \)
19 \( 1 + 1.35T + 19T^{2} \)
23 \( 1 - 3.98T + 23T^{2} \)
29 \( 1 - 3.50T + 29T^{2} \)
31 \( 1 - 4.07T + 31T^{2} \)
37 \( 1 + 7.33T + 37T^{2} \)
41 \( 1 + 7.61T + 41T^{2} \)
43 \( 1 - 3.57T + 43T^{2} \)
47 \( 1 + 0.864T + 47T^{2} \)
53 \( 1 + 4.84T + 53T^{2} \)
59 \( 1 - 2.86T + 59T^{2} \)
61 \( 1 - 8.17T + 61T^{2} \)
67 \( 1 - 6.80T + 67T^{2} \)
71 \( 1 + 1.78T + 71T^{2} \)
73 \( 1 - 11.6T + 73T^{2} \)
79 \( 1 + 9.39T + 79T^{2} \)
83 \( 1 - 6.30T + 83T^{2} \)
89 \( 1 - 3.25T + 89T^{2} \)
97 \( 1 - 4.97T + 97T^{2} \)
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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

−7.40937477945706394065052071674, −6.72225307561713136376172996503, −6.20966055721093667138598929283, −5.35777282747358616733203275717, −4.88246907263921079404536023943, −3.89293937498383354211729065769, −3.13254238592372228760152326835, −2.07424154909793939839053541953, −0.951508851710218584946822495313, 0, 0.951508851710218584946822495313, 2.07424154909793939839053541953, 3.13254238592372228760152326835, 3.89293937498383354211729065769, 4.88246907263921079404536023943, 5.35777282747358616733203275717, 6.20966055721093667138598929283, 6.72225307561713136376172996503, 7.40937477945706394065052071674

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