Properties

Label 2-13e2-169.12-c1-0-4
Degree $2$
Conductor $169$
Sign $0.934 + 0.356i$
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  + (−1.24 + 0.862i)2-s + (−1.71 − 0.898i)3-s + (0.107 − 0.284i)4-s + (0.371 + 0.418i)5-s + (2.91 − 0.353i)6-s + (−0.356 − 1.44i)7-s + (−0.615 − 2.49i)8-s + (0.418 + 0.606i)9-s + (−0.824 − 0.203i)10-s + (3.96 + 2.73i)11-s + (−0.440 + 0.390i)12-s + (2.06 − 2.95i)13-s + (1.69 + 1.49i)14-s + (−0.258 − 1.05i)15-s + (3.38 + 2.99i)16-s + (7.61 − 1.87i)17-s + ⋯
L(s)  = 1  + (−0.883 + 0.609i)2-s + (−0.988 − 0.518i)3-s + (0.0539 − 0.142i)4-s + (0.165 + 0.187i)5-s + (1.18 − 0.144i)6-s + (−0.134 − 0.546i)7-s + (−0.217 − 0.883i)8-s + (0.139 + 0.202i)9-s + (−0.260 − 0.0642i)10-s + (1.19 + 0.825i)11-s + (−0.127 + 0.112i)12-s + (0.572 − 0.819i)13-s + (0.451 + 0.400i)14-s + (−0.0668 − 0.271i)15-s + (0.845 + 0.748i)16-s + (1.84 − 0.455i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.934 + 0.356i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.934 + 0.356i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(169\)    =    \(13^{2}\)
Sign: $0.934 + 0.356i$
Analytic conductor: \(1.34947\)
Root analytic conductor: \(1.16166\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{169} (12, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 169,\ (\ :1/2),\ 0.934 + 0.356i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.517235 - 0.0954361i\)
\(L(\frac12)\) \(\approx\) \(0.517235 - 0.0954361i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 + (-2.06 + 2.95i)T \)
good2 \( 1 + (1.24 - 0.862i)T + (0.709 - 1.87i)T^{2} \)
3 \( 1 + (1.71 + 0.898i)T + (1.70 + 2.46i)T^{2} \)
5 \( 1 + (-0.371 - 0.418i)T + (-0.602 + 4.96i)T^{2} \)
7 \( 1 + (0.356 + 1.44i)T + (-6.19 + 3.25i)T^{2} \)
11 \( 1 + (-3.96 - 2.73i)T + (3.90 + 10.2i)T^{2} \)
17 \( 1 + (-7.61 + 1.87i)T + (15.0 - 7.90i)T^{2} \)
19 \( 1 + 5.51iT - 19T^{2} \)
23 \( 1 + 6.37T + 23T^{2} \)
29 \( 1 + (-1.15 - 1.66i)T + (-10.2 + 27.1i)T^{2} \)
31 \( 1 + (3.06 - 0.371i)T + (30.0 - 7.41i)T^{2} \)
37 \( 1 + (6.28 - 0.763i)T + (35.9 - 8.85i)T^{2} \)
41 \( 1 + (1.57 - 3.00i)T + (-23.2 - 33.7i)T^{2} \)
43 \( 1 + (-1.14 + 9.46i)T + (-41.7 - 10.2i)T^{2} \)
47 \( 1 + (-0.734 + 0.278i)T + (35.1 - 31.1i)T^{2} \)
53 \( 1 + (-9.64 + 2.37i)T + (46.9 - 24.6i)T^{2} \)
59 \( 1 + (0.709 + 0.801i)T + (-7.11 + 58.5i)T^{2} \)
61 \( 1 + (-5.70 - 1.40i)T + (54.0 + 28.3i)T^{2} \)
67 \( 1 + (1.44 - 0.546i)T + (50.1 - 44.4i)T^{2} \)
71 \( 1 + (2.49 - 4.74i)T + (-40.3 - 58.4i)T^{2} \)
73 \( 1 + (-4.52 - 3.12i)T + (25.8 + 68.2i)T^{2} \)
79 \( 1 + (-0.523 - 1.38i)T + (-59.1 + 52.3i)T^{2} \)
83 \( 1 + (1.23 + 2.34i)T + (-47.1 + 68.3i)T^{2} \)
89 \( 1 - 15.8iT - 89T^{2} \)
97 \( 1 + (2.04 - 2.31i)T + (-11.6 - 96.2i)T^{2} \)
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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

−12.39165191597925482299650382581, −11.94034684452332314368856179996, −10.49769425240098237246470242201, −9.721201659593743982668220540705, −8.522311706477523525085718590169, −7.25931124863505315204374969023, −6.72354588966592235893779224794, −5.58894843469745669100044903416, −3.72369989235150067782020060701, −0.863033503767424180938430195703, 1.47769798383720707806711446226, 3.76017717468758165641695222094, 5.58401271475612127489021710258, 6.02579794217574104307954208541, 8.101246287300740133670751785503, 9.080648630167087161099221854173, 9.944176054609957687368788858966, 10.75003590255133247699504392123, 11.79744708494983002936438790468, 12.07761520485225860178314869094

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