Properties

Label 2-13e2-13.4-c3-0-23
Degree $2$
Conductor $169$
Sign $0.000427 + 0.999i$
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  + (2.73 + 1.58i)2-s + (−3.54 + 6.13i)3-s + (0.997 + 1.72i)4-s − 13.6i·5-s + (−19.4 + 11.2i)6-s + (−12.4 + 7.16i)7-s − 18.9i·8-s + (−11.6 − 20.1i)9-s + (21.5 − 37.2i)10-s + (−58.6 − 33.8i)11-s − 14.1·12-s − 45.3·14-s + (83.5 + 48.2i)15-s + (37.9 − 65.7i)16-s + (0.168 + 0.292i)17-s − 73.5i·18-s + ⋯
L(s)  = 1  + (0.967 + 0.558i)2-s + (−0.682 + 1.18i)3-s + (0.124 + 0.215i)4-s − 1.21i·5-s + (−1.32 + 0.762i)6-s + (−0.670 + 0.386i)7-s − 0.839i·8-s + (−0.430 − 0.745i)9-s + (0.679 − 1.17i)10-s + (−1.60 − 0.928i)11-s − 0.340·12-s − 0.864·14-s + (1.43 + 0.829i)15-s + (0.593 − 1.02i)16-s + (0.00240 + 0.00417i)17-s − 0.962i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.000427 + 0.999i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.000427 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(169\)    =    \(13^{2}\)
Sign: $0.000427 + 0.999i$
Analytic conductor: \(9.97132\)
Root analytic conductor: \(3.15774\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{169} (147, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 169,\ (\ :3/2),\ 0.000427 + 0.999i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.475348 - 0.475145i\)
\(L(\frac12)\) \(\approx\) \(0.475348 - 0.475145i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-2.73 - 1.58i)T + (4 + 6.92i)T^{2} \)
3 \( 1 + (3.54 - 6.13i)T + (-13.5 - 23.3i)T^{2} \)
5 \( 1 + 13.6iT - 125T^{2} \)
7 \( 1 + (12.4 - 7.16i)T + (171.5 - 297. i)T^{2} \)
11 \( 1 + (58.6 + 33.8i)T + (665.5 + 1.15e3i)T^{2} \)
17 \( 1 + (-0.168 - 0.292i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-35.0 + 20.2i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (77.8 - 134. i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-16.8 + 29.2i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + 157. iT - 2.97e4T^{2} \)
37 \( 1 + (50.7 + 29.3i)T + (2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (51.3 + 29.6i)T + (3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (104. + 180. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 - 221. iT - 1.03e5T^{2} \)
53 \( 1 + 409.T + 1.48e5T^{2} \)
59 \( 1 + (150. - 86.7i)T + (1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (280. + 485. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-233. - 134. i)T + (1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + (-52.8 + 30.4i)T + (1.78e5 - 3.09e5i)T^{2} \)
73 \( 1 + 282. iT - 3.89e5T^{2} \)
79 \( 1 - 984.T + 4.93e5T^{2} \)
83 \( 1 - 1.20e3iT - 5.71e5T^{2} \)
89 \( 1 + (467. + 269. i)T + (3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (-1.37e3 + 793. i)T + (4.56e5 - 7.90e5i)T^{2} \)
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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

−12.33855665925114857598263811034, −11.11399563109332611298704096367, −9.970350735779734734514386709921, −9.276510639825065578425324099390, −7.86632663000566043494621341992, −5.99029226876575462410722833414, −5.40068758496163101544705075126, −4.69507157471407941699287656090, −3.43033810792481412298909373334, −0.23157305196068609620612084422, 2.19618177261270112676183898133, 3.25326628942087544760016843813, 4.93264971215540419798124333510, 6.21210232773386062090339091860, 7.09243243992721683063502418070, 7.991161883263513601743431016855, 10.16355034218125278153183975629, 10.79526693633634944304665242014, 11.92147246805893273526592005432, 12.63069400911078532046723780693

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