Properties

Label 2-13e2-1.1-c1-0-4
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  + 2.24·2-s − 0.554·3-s + 3.04·4-s + 1.44·5-s − 1.24·6-s − 2.04·7-s + 2.35·8-s − 2.69·9-s + 3.24·10-s + 2.55·11-s − 1.69·12-s − 4.60·14-s − 0.801·15-s − 0.801·16-s − 5.29·17-s − 6.04·18-s + 5.85·19-s + 4.40·20-s + 1.13·21-s + 5.74·22-s − 1.89·23-s − 1.30·24-s − 2.91·25-s + 3.15·27-s − 6.24·28-s + 2.26·29-s − 1.80·30-s + ⋯
L(s)  = 1  + 1.58·2-s − 0.320·3-s + 1.52·4-s + 0.646·5-s − 0.509·6-s − 0.774·7-s + 0.833·8-s − 0.897·9-s + 1.02·10-s + 0.770·11-s − 0.488·12-s − 1.23·14-s − 0.207·15-s − 0.200·16-s − 1.28·17-s − 1.42·18-s + 1.34·19-s + 0.985·20-s + 0.248·21-s + 1.22·22-s − 0.394·23-s − 0.266·24-s − 0.582·25-s + 0.607·27-s − 1.18·28-s + 0.421·29-s − 0.328·30-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

\(L(1)\) \(\approx\) \(2.240862488\)
\(L(\frac12)\) \(\approx\) \(2.240862488\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.24T + 2T^{2} \)
3 \( 1 + 0.554T + 3T^{2} \)
5 \( 1 - 1.44T + 5T^{2} \)
7 \( 1 + 2.04T + 7T^{2} \)
11 \( 1 - 2.55T + 11T^{2} \)
17 \( 1 + 5.29T + 17T^{2} \)
19 \( 1 - 5.85T + 19T^{2} \)
23 \( 1 + 1.89T + 23T^{2} \)
29 \( 1 - 2.26T + 29T^{2} \)
31 \( 1 - 4.26T + 31T^{2} \)
37 \( 1 + 5.35T + 37T^{2} \)
41 \( 1 + 1.27T + 41T^{2} \)
43 \( 1 - 6.13T + 43T^{2} \)
47 \( 1 - 2.95T + 47T^{2} \)
53 \( 1 - 5.52T + 53T^{2} \)
59 \( 1 - 12.2T + 59T^{2} \)
61 \( 1 - 8.56T + 61T^{2} \)
67 \( 1 + 0.576T + 67T^{2} \)
71 \( 1 - 4.59T + 71T^{2} \)
73 \( 1 - 10.5T + 73T^{2} \)
79 \( 1 + 15.7T + 79T^{2} \)
83 \( 1 + 7.72T + 83T^{2} \)
89 \( 1 + 6.61T + 89T^{2} \)
97 \( 1 + 11.9T + 97T^{2} \)
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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

−12.94081045019636767147359476365, −11.92674209644277126349347180296, −11.33150064057715265287341268911, −9.921099931600111742647015868312, −8.816353062533442556770931127476, −6.87114205305089699269934736758, −6.11296835528273051749193674268, −5.26167575173753922282991782557, −3.86542499175143892522610473183, −2.58504684420432331672462752832, 2.58504684420432331672462752832, 3.86542499175143892522610473183, 5.26167575173753922282991782557, 6.11296835528273051749193674268, 6.87114205305089699269934736758, 8.816353062533442556770931127476, 9.921099931600111742647015868312, 11.33150064057715265287341268911, 11.92674209644277126349347180296, 12.94081045019636767147359476365

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