Properties

Label 2-8047-1.1-c1-0-535
Degree $2$
Conductor $8047$
Sign $-1$
Analytic cond. $64.2556$
Root an. cond. $8.01596$
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  − 2.45·2-s + 2.98·3-s + 4.04·4-s − 1.52·5-s − 7.34·6-s + 5.18·7-s − 5.01·8-s + 5.92·9-s + 3.75·10-s − 3.38·11-s + 12.0·12-s + 13-s − 12.7·14-s − 4.56·15-s + 4.24·16-s − 6.38·17-s − 14.5·18-s − 0.342·19-s − 6.17·20-s + 15.4·21-s + 8.33·22-s − 2.04·23-s − 14.9·24-s − 2.66·25-s − 2.45·26-s + 8.72·27-s + 20.9·28-s + ⋯
L(s)  = 1  − 1.73·2-s + 1.72·3-s + 2.02·4-s − 0.683·5-s − 2.99·6-s + 1.95·7-s − 1.77·8-s + 1.97·9-s + 1.18·10-s − 1.02·11-s + 3.48·12-s + 0.277·13-s − 3.40·14-s − 1.17·15-s + 1.06·16-s − 1.54·17-s − 3.43·18-s − 0.0785·19-s − 1.37·20-s + 3.37·21-s + 1.77·22-s − 0.425·23-s − 3.05·24-s − 0.533·25-s − 0.481·26-s + 1.67·27-s + 3.95·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8047 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8047 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8047\)    =    \(13 \cdot 619\)
Sign: $-1$
Analytic conductor: \(64.2556\)
Root analytic conductor: \(8.01596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 8047,\ (\ :1/2),\ -1)\)

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)$
bad13 \( 1 - T \)
619 \( 1 - T \)
good2 \( 1 + 2.45T + 2T^{2} \)
3 \( 1 - 2.98T + 3T^{2} \)
5 \( 1 + 1.52T + 5T^{2} \)
7 \( 1 - 5.18T + 7T^{2} \)
11 \( 1 + 3.38T + 11T^{2} \)
17 \( 1 + 6.38T + 17T^{2} \)
19 \( 1 + 0.342T + 19T^{2} \)
23 \( 1 + 2.04T + 23T^{2} \)
29 \( 1 + 10.3T + 29T^{2} \)
31 \( 1 - 0.408T + 31T^{2} \)
37 \( 1 + 6.11T + 37T^{2} \)
41 \( 1 - 7.78T + 41T^{2} \)
43 \( 1 + 5.88T + 43T^{2} \)
47 \( 1 + 7.49T + 47T^{2} \)
53 \( 1 - 5.42T + 53T^{2} \)
59 \( 1 + 11.6T + 59T^{2} \)
61 \( 1 + 3.06T + 61T^{2} \)
67 \( 1 - 12.7T + 67T^{2} \)
71 \( 1 - 4.94T + 71T^{2} \)
73 \( 1 + 5.49T + 73T^{2} \)
79 \( 1 - 3.51T + 79T^{2} \)
83 \( 1 - 10.9T + 83T^{2} \)
89 \( 1 + 7.93T + 89T^{2} \)
97 \( 1 + 14.5T + 97T^{2} \)
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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

−7.69341307684410177077536443726, −7.56591358866770320041335367142, −6.68569944326868509607165933973, −5.34599163142532319994472958180, −4.41377373646595331564183472905, −3.75686057599269068300232783278, −2.57533909226168964656650270512, −2.01594638448538686140630148878, −1.52435860824979313432587944173, 0, 1.52435860824979313432587944173, 2.01594638448538686140630148878, 2.57533909226168964656650270512, 3.75686057599269068300232783278, 4.41377373646595331564183472905, 5.34599163142532319994472958180, 6.68569944326868509607165933973, 7.56591358866770320041335367142, 7.69341307684410177077536443726

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