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Properties

Label 2-6137-1.1-c1-0-317
Degree $2$
Conductor $6137$
Sign $-1$
Analytic cond. $49.0041$
Root an. cond. $7.00029$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 4-s − 2·5-s + 4·7-s − 3·8-s − 3·9-s − 2·10-s + 2·13-s + 4·14-s − 16-s + 17-s − 3·18-s + 2·20-s + 4·23-s − 25-s + 2·26-s − 4·28-s − 6·29-s − 4·31-s + 5·32-s + 34-s − 8·35-s + 3·36-s + 2·37-s + 6·40-s + 6·41-s + 4·43-s + ⋯
L(s)  = 1  + 0.707·2-s − 1/2·4-s − 0.894·5-s + 1.51·7-s − 1.06·8-s − 9-s − 0.632·10-s + 0.554·13-s + 1.06·14-s − 1/4·16-s + 0.242·17-s − 0.707·18-s + 0.447·20-s + 0.834·23-s − 1/5·25-s + 0.392·26-s − 0.755·28-s − 1.11·29-s − 0.718·31-s + 0.883·32-s + 0.171·34-s − 1.35·35-s + 1/2·36-s + 0.328·37-s + 0.948·40-s + 0.937·41-s + 0.609·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6137 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6137 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6137\)    =    \(17 \cdot 19^{2}\)
Sign: $-1$
Analytic conductor: \(49.0041\)
Root analytic conductor: \(7.00029\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 6137,\ (\ :1/2),\ -1)\)

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)$
bad17 \( 1 - T \)
19 \( 1 \)
good2 \( 1 - T + p T^{2} \)
3 \( 1 + p T^{2} \)
5 \( 1 + 2 T + p T^{2} \)
7 \( 1 - 4 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 4 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + 12 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−7.85987780234322795046014866090, −7.11612784029660068549214102487, −5.86260696459739413423436350311, −5.54363232495152288696215311374, −4.72756278877742608574999039026, −4.12014827255109957812466528240, −3.46394875273714724608087415298, −2.55952301596916907351467871473, −1.27552475033395022127866657387, 0, 1.27552475033395022127866657387, 2.55952301596916907351467871473, 3.46394875273714724608087415298, 4.12014827255109957812466528240, 4.72756278877742608574999039026, 5.54363232495152288696215311374, 5.86260696459739413423436350311, 7.11612784029660068549214102487, 7.85987780234322795046014866090

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