Properties

Label 2-2695-1.1-c1-0-98
Degree $2$
Conductor $2695$
Sign $-1$
Analytic cond. $21.5196$
Root an. cond. $4.63893$
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  − 2.15·2-s + 2.37·3-s + 2.63·4-s − 5-s − 5.11·6-s − 1.36·8-s + 2.64·9-s + 2.15·10-s − 11-s + 6.25·12-s − 5.45·13-s − 2.37·15-s − 2.33·16-s − 0.865·17-s − 5.69·18-s + 2.98·19-s − 2.63·20-s + 2.15·22-s + 6.14·23-s − 3.23·24-s + 25-s + 11.7·26-s − 0.837·27-s − 0.799·29-s + 5.11·30-s − 1.38·31-s + 7.74·32-s + ⋯
L(s)  = 1  − 1.52·2-s + 1.37·3-s + 1.31·4-s − 0.447·5-s − 2.08·6-s − 0.481·8-s + 0.882·9-s + 0.680·10-s − 0.301·11-s + 1.80·12-s − 1.51·13-s − 0.613·15-s − 0.584·16-s − 0.209·17-s − 1.34·18-s + 0.685·19-s − 0.588·20-s + 0.458·22-s + 1.28·23-s − 0.659·24-s + 0.200·25-s + 2.30·26-s − 0.161·27-s − 0.148·29-s + 0.933·30-s − 0.249·31-s + 1.36·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2695 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2695 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2695\)    =    \(5 \cdot 7^{2} \cdot 11\)
Sign: $-1$
Analytic conductor: \(21.5196\)
Root analytic conductor: \(4.63893\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 2695,\ (\ :1/2),\ -1)\)

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 \( 1 \)
11 \( 1 + T \)
good2 \( 1 + 2.15T + 2T^{2} \)
3 \( 1 - 2.37T + 3T^{2} \)
13 \( 1 + 5.45T + 13T^{2} \)
17 \( 1 + 0.865T + 17T^{2} \)
19 \( 1 - 2.98T + 19T^{2} \)
23 \( 1 - 6.14T + 23T^{2} \)
29 \( 1 + 0.799T + 29T^{2} \)
31 \( 1 + 1.38T + 31T^{2} \)
37 \( 1 - 7.02T + 37T^{2} \)
41 \( 1 + 11.3T + 41T^{2} \)
43 \( 1 + 1.49T + 43T^{2} \)
47 \( 1 + 3.54T + 47T^{2} \)
53 \( 1 + 8.30T + 53T^{2} \)
59 \( 1 - 5.74T + 59T^{2} \)
61 \( 1 - 3.97T + 61T^{2} \)
67 \( 1 - 1.46T + 67T^{2} \)
71 \( 1 + 1.38T + 71T^{2} \)
73 \( 1 + 1.27T + 73T^{2} \)
79 \( 1 + 13.8T + 79T^{2} \)
83 \( 1 + 16.7T + 83T^{2} \)
89 \( 1 + 18.4T + 89T^{2} \)
97 \( 1 - 2.46T + 97T^{2} \)
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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

−8.448562692624902645510478758710, −7.966100310475824317491032214628, −7.27251947665458201101903536126, −6.87513386151300175931244600979, −5.27463115404193944422899031907, −4.36640168315260478110140593162, −3.11760342843965074779477869619, −2.54179993955079052857380990725, −1.48398293947217292054912099066, 0, 1.48398293947217292054912099066, 2.54179993955079052857380990725, 3.11760342843965074779477869619, 4.36640168315260478110140593162, 5.27463115404193944422899031907, 6.87513386151300175931244600979, 7.27251947665458201101903536126, 7.966100310475824317491032214628, 8.448562692624902645510478758710

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