Properties

Label 2-13e2-1.1-c1-0-1
Degree $2$
Conductor $169$
Sign $1$
Analytic cond. $1.34947$
Root an. cond. $1.16166$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.73·2-s + 2·3-s + 0.999·4-s + 1.73·5-s − 3.46·6-s + 1.73·8-s + 9-s − 2.99·10-s + 1.99·12-s + 3.46·15-s − 5·16-s + 3·17-s − 1.73·18-s + 3.46·19-s + 1.73·20-s + 6·23-s + 3.46·24-s − 2.00·25-s − 4·27-s + 3·29-s − 5.99·30-s − 3.46·31-s + 5.19·32-s − 5.19·34-s + 0.999·36-s − 8.66·37-s − 5.99·38-s + ⋯
L(s)  = 1  − 1.22·2-s + 1.15·3-s + 0.499·4-s + 0.774·5-s − 1.41·6-s + 0.612·8-s + 0.333·9-s − 0.948·10-s + 0.577·12-s + 0.894·15-s − 1.25·16-s + 0.727·17-s − 0.408·18-s + 0.794·19-s + 0.387·20-s + 1.25·23-s + 0.707·24-s − 0.400·25-s − 0.769·27-s + 0.557·29-s − 1.09·30-s − 0.622·31-s + 0.918·32-s − 0.891·34-s + 0.166·36-s − 1.42·37-s − 0.973·38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(169\)    =    \(13^{2}\)
Sign: $1$
Analytic conductor: \(1.34947\)
Root analytic conductor: \(1.16166\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 169,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9663863038\)
\(L(\frac12)\) \(\approx\) \(0.9663863038\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
good2 \( 1 + 1.73T + 2T^{2} \)
3 \( 1 - 2T + 3T^{2} \)
5 \( 1 - 1.73T + 5T^{2} \)
7 \( 1 + 7T^{2} \)
11 \( 1 + 11T^{2} \)
17 \( 1 - 3T + 17T^{2} \)
19 \( 1 - 3.46T + 19T^{2} \)
23 \( 1 - 6T + 23T^{2} \)
29 \( 1 - 3T + 29T^{2} \)
31 \( 1 + 3.46T + 31T^{2} \)
37 \( 1 + 8.66T + 37T^{2} \)
41 \( 1 + 5.19T + 41T^{2} \)
43 \( 1 + 8T + 43T^{2} \)
47 \( 1 + 3.46T + 47T^{2} \)
53 \( 1 + 3T + 53T^{2} \)
59 \( 1 - 6.92T + 59T^{2} \)
61 \( 1 - T + 61T^{2} \)
67 \( 1 - 3.46T + 67T^{2} \)
71 \( 1 + 3.46T + 71T^{2} \)
73 \( 1 - 1.73T + 73T^{2} \)
79 \( 1 - 4T + 79T^{2} \)
83 \( 1 + 13.8T + 83T^{2} \)
89 \( 1 - 6.92T + 89T^{2} \)
97 \( 1 + 6.92T + 97T^{2} \)
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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

−13.07149786603394019910645113250, −11.52515822423401509812583099945, −10.23881727093134391009984706507, −9.588508745175309869068694658006, −8.800662547309621166921013579389, −7.999777026071579701241342315621, −6.94378083482087772843213542582, −5.21823114907427806413664401822, −3.26403351690720069103565525424, −1.71909036867686537100924621887, 1.71909036867686537100924621887, 3.26403351690720069103565525424, 5.21823114907427806413664401822, 6.94378083482087772843213542582, 7.999777026071579701241342315621, 8.800662547309621166921013579389, 9.588508745175309869068694658006, 10.23881727093134391009984706507, 11.52515822423401509812583099945, 13.07149786603394019910645113250

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