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Properties

Label 2-13e2-13.9-c1-0-1
Degree $2$
Conductor $169$
Sign $-0.434 - 0.900i$
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  + (−1.12 + 1.94i)2-s + (0.277 − 0.480i)3-s + (−1.52 − 2.64i)4-s + 1.44·5-s + (0.623 + 1.07i)6-s + (1.02 + 1.77i)7-s + 2.35·8-s + (1.34 + 2.33i)9-s + (−1.62 + 2.81i)10-s + (−1.27 + 2.21i)11-s − 1.69·12-s − 4.60·14-s + (0.400 − 0.694i)15-s + (0.400 − 0.694i)16-s + (2.64 + 4.58i)17-s − 6.04·18-s + ⋯
L(s)  = 1  + (−0.794 + 1.37i)2-s + (0.160 − 0.277i)3-s + (−0.762 − 1.32i)4-s + 0.646·5-s + (0.254 + 0.440i)6-s + (0.387 + 0.670i)7-s + 0.833·8-s + (0.448 + 0.777i)9-s + (−0.513 + 0.889i)10-s + (−0.385 + 0.667i)11-s − 0.488·12-s − 1.23·14-s + (0.103 − 0.179i)15-s + (0.100 − 0.173i)16-s + (0.642 + 1.11i)17-s − 1.42·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.434 - 0.900i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.434 - 0.900i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(169\)    =    \(13^{2}\)
Sign: $-0.434 - 0.900i$
Analytic conductor: \(1.34947\)
Root analytic conductor: \(1.16166\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{169} (22, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 169,\ (\ :1/2),\ -0.434 - 0.900i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.468862 + 0.746537i\)
\(L(\frac12)\) \(\approx\) \(0.468862 + 0.746537i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.12 - 1.94i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (-0.277 + 0.480i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 - 1.44T + 5T^{2} \)
7 \( 1 + (-1.02 - 1.77i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.27 - 2.21i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-2.64 - 4.58i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.92 + 5.06i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.945 + 1.63i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.13 - 1.96i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 4.26T + 31T^{2} \)
37 \( 1 + (-2.67 + 4.63i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-0.637 + 1.10i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.06 + 5.31i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 2.95T + 47T^{2} \)
53 \( 1 - 5.52T + 53T^{2} \)
59 \( 1 + (6.10 + 10.5i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.28 + 7.41i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.288 + 0.499i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (2.29 + 3.97i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 10.5T + 73T^{2} \)
79 \( 1 + 15.7T + 79T^{2} \)
83 \( 1 + 7.72T + 83T^{2} \)
89 \( 1 + (-3.30 + 5.72i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-5.96 - 10.3i)T + (-48.5 + 84.0i)T^{2} \)
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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

−13.30274802493991943642062769430, −12.34513401850382240147921692645, −10.70270267797774437592931441116, −9.789333694698740747997270509197, −8.734142681309732121573547923881, −7.945879412669335960905066978024, −6.98069586484077288246712179424, −5.88007385283961264644554586828, −4.87754023786801768500699109148, −2.10733452583127623776427609752, 1.21457931513965198247650443666, 2.92403365456373249923848739359, 4.17350699102445290711291341779, 5.98004456164209953798977206940, 7.66314262783182347160004044771, 8.773825595905640774039793309036, 9.854957865324813564497159491176, 10.19863433196619759390761616131, 11.33204111691160329989223093198, 12.15836106979494755575747082313

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