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Properties

Label 2-13e2-13.3-c3-0-21
Degree $2$
Conductor $169$
Sign $0.859 + 0.511i$
Analytic cond. $9.97132$
Root an. cond. $3.15774$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2 + 3.46i)2-s + (−1 − 1.73i)3-s + (−3.99 + 6.92i)4-s − 17·5-s + (3.99 − 6.92i)6-s + (10 − 17.3i)7-s + (11.5 − 19.9i)9-s + (−34 − 58.8i)10-s + (−16 − 27.7i)11-s + 15.9·12-s + 80·14-s + (17 + 29.4i)15-s + (31.9 + 55.4i)16-s + (6.5 − 11.2i)17-s + 92·18-s + (15 − 25.9i)19-s + ⋯
L(s)  = 1  + (0.707 + 1.22i)2-s + (−0.192 − 0.333i)3-s + (−0.499 + 0.866i)4-s − 1.52·5-s + (0.272 − 0.471i)6-s + (0.539 − 0.935i)7-s + (0.425 − 0.737i)9-s + (−1.07 − 1.86i)10-s + (−0.438 − 0.759i)11-s + 0.384·12-s + 1.52·14-s + (0.292 + 0.506i)15-s + (0.499 + 0.866i)16-s + (0.0927 − 0.160i)17-s + 1.20·18-s + (0.181 − 0.313i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.859 + 0.511i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.859 + 0.511i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(169\)    =    \(13^{2}\)
Sign: $0.859 + 0.511i$
Analytic conductor: \(9.97132\)
Root analytic conductor: \(3.15774\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{169} (146, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 169,\ (\ :3/2),\ 0.859 + 0.511i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.49639 - 0.411260i\)
\(L(\frac12)\) \(\approx\) \(1.49639 - 0.411260i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-2 - 3.46i)T + (-4 + 6.92i)T^{2} \)
3 \( 1 + (1 + 1.73i)T + (-13.5 + 23.3i)T^{2} \)
5 \( 1 + 17T + 125T^{2} \)
7 \( 1 + (-10 + 17.3i)T + (-171.5 - 297. i)T^{2} \)
11 \( 1 + (16 + 27.7i)T + (-665.5 + 1.15e3i)T^{2} \)
17 \( 1 + (-6.5 + 11.2i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-15 + 25.9i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (39 + 67.5i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (98.5 + 170. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 - 74T + 2.97e4T^{2} \)
37 \( 1 + (113.5 + 196. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (82.5 + 142. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-78 + 135. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 - 162T + 1.03e5T^{2} \)
53 \( 1 - 93T + 1.48e5T^{2} \)
59 \( 1 + (432 - 748. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (72.5 - 125. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-431 - 746. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + (-327 + 566. i)T + (-1.78e5 - 3.09e5i)T^{2} \)
73 \( 1 + 215T + 3.89e5T^{2} \)
79 \( 1 + 76T + 4.93e5T^{2} \)
83 \( 1 + 628T + 5.71e5T^{2} \)
89 \( 1 + (133 + 230. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (-119 + 206. i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−12.39176177549082977324248987549, −11.44693017992065744608299047222, −10.48045563606500048217952498876, −8.574564944770912953903147810920, −7.56976360833430144025817501044, −7.18058417806014843596355859254, −5.89460402004959842144283740489, −4.46567923090857713114237797257, −3.75138284507640079597406899050, −0.60756928641029764695474985099, 1.85515376064640039874511519575, 3.35636653765253081209316927232, 4.50277853019225456721620594290, 5.21329043516230545724610787366, 7.42467951987553773496092091748, 8.216002573546574347883330483928, 9.807876918805277224929245523997, 10.84851109072130531679148159636, 11.51825986183605080674148060342, 12.23327519901452037253715018860

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