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Properties

Label 2-13e2-13.12-c1-0-1
Degree $2$
Conductor $169$
Sign $-0.246 - 0.969i$
Analytic cond. $1.34947$
Root an. cond. $1.16166$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.801i·2-s − 2.24·3-s + 1.35·4-s + 0.246i·5-s − 1.80i·6-s + 2.35i·7-s + 2.69i·8-s + 2.04·9-s − 0.198·10-s + 4.24i·11-s − 3.04·12-s − 1.89·14-s − 0.554i·15-s + 0.554·16-s − 2.15·17-s + 1.64i·18-s + ⋯
L(s)  = 1  + 0.567i·2-s − 1.29·3-s + 0.678·4-s + 0.110i·5-s − 0.735i·6-s + 0.890i·7-s + 0.951i·8-s + 0.682·9-s − 0.0626·10-s + 1.28i·11-s − 0.880·12-s − 0.505·14-s − 0.143i·15-s + 0.138·16-s − 0.523·17-s + 0.387i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.246 - 0.969i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.246 - 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(169\)    =    \(13^{2}\)
Sign: $-0.246 - 0.969i$
Analytic conductor: \(1.34947\)
Root analytic conductor: \(1.16166\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{169} (168, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 169,\ (\ :1/2),\ -0.246 - 0.969i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.526260 + 0.677133i\)
\(L(\frac12)\) \(\approx\) \(0.526260 + 0.677133i\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.801iT - 2T^{2} \)
3 \( 1 + 2.24T + 3T^{2} \)
5 \( 1 - 0.246iT - 5T^{2} \)
7 \( 1 - 2.35iT - 7T^{2} \)
11 \( 1 - 4.24iT - 11T^{2} \)
17 \( 1 + 2.15T + 17T^{2} \)
19 \( 1 + 0.0881iT - 19T^{2} \)
23 \( 1 + 1.49T + 23T^{2} \)
29 \( 1 - 4.63T + 29T^{2} \)
31 \( 1 + 6.63iT - 31T^{2} \)
37 \( 1 + 5.69iT - 37T^{2} \)
41 \( 1 + 11.5iT - 41T^{2} \)
43 \( 1 - 0.295T + 43T^{2} \)
47 \( 1 - 7.35iT - 47T^{2} \)
53 \( 1 + 10.3T + 53T^{2} \)
59 \( 1 - 6.78iT - 59T^{2} \)
61 \( 1 - 3.47T + 61T^{2} \)
67 \( 1 - 7.67iT - 67T^{2} \)
71 \( 1 + 8.66iT - 71T^{2} \)
73 \( 1 + 6.73iT - 73T^{2} \)
79 \( 1 - 9.97T + 79T^{2} \)
83 \( 1 - 1.60iT - 83T^{2} \)
89 \( 1 - 2.88iT - 89T^{2} \)
97 \( 1 + 8.05iT - 97T^{2} \)
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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

−12.60292939541458750400440328164, −12.07372140353207185586078077439, −11.19309343562318784894356375339, −10.35833148982552036260034672881, −8.960870351012221547408041379622, −7.53302451012761119921686975896, −6.56928182405638460721847415966, −5.76620790941156845177220617520, −4.74988371678695169251375514126, −2.31525938369508570616862188854, 0.972677609427910568972927347823, 3.23468304526772147905274936571, 4.81365681898570112082128468052, 6.21803460942848297855065815726, 6.85475543179598805055419457533, 8.364029985393972812836688473130, 10.03523391941710643807817071185, 10.84138419094814527551913166934, 11.31134897397206640333604955449, 12.20506584220473222317236584723

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