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Properties

Label 2-13e2-13.3-c3-0-16
Degree $2$
Conductor $169$
Sign $0.434 - 0.900i$
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  + (2.41 + 4.17i)2-s + (−2.22 − 3.84i)3-s + (−7.64 + 13.2i)4-s + 12.7·5-s + (10.7 − 18.5i)6-s + (13.0 − 22.6i)7-s − 35.1·8-s + (3.62 − 6.28i)9-s + (30.8 + 53.3i)10-s + (21.1 + 36.6i)11-s + 67.9·12-s + 126.·14-s + (−28.3 − 49.1i)15-s + (−23.6 − 41.0i)16-s + (13.6 − 23.6i)17-s + 35.0·18-s + ⋯
L(s)  = 1  + (0.853 + 1.47i)2-s + (−0.427 − 0.740i)3-s + (−0.955 + 1.65i)4-s + 1.14·5-s + (0.729 − 1.26i)6-s + (0.706 − 1.22i)7-s − 1.55·8-s + (0.134 − 0.232i)9-s + (0.974 + 1.68i)10-s + (0.580 + 1.00i)11-s + 1.63·12-s + 2.41·14-s + (−0.488 − 0.845i)15-s + (−0.370 − 0.641i)16-s + (0.194 − 0.337i)17-s + 0.458·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.434 - 0.900i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.434 - 0.900i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(169\)    =    \(13^{2}\)
Sign: $0.434 - 0.900i$
Analytic conductor: \(9.97132\)
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: \((2,\ 169,\ (\ :3/2),\ 0.434 - 0.900i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.47175 + 1.55238i\)
\(L(\frac12)\) \(\approx\) \(2.47175 + 1.55238i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-2.41 - 4.17i)T + (-4 + 6.92i)T^{2} \)
3 \( 1 + (2.22 + 3.84i)T + (-13.5 + 23.3i)T^{2} \)
5 \( 1 - 12.7T + 125T^{2} \)
7 \( 1 + (-13.0 + 22.6i)T + (-171.5 - 297. i)T^{2} \)
11 \( 1 + (-21.1 - 36.6i)T + (-665.5 + 1.15e3i)T^{2} \)
17 \( 1 + (-13.6 + 23.6i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (6.55 - 11.3i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-14.3 - 24.9i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-70.8 - 122. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 - 56.0T + 2.97e4T^{2} \)
37 \( 1 + (156. + 271. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (-176. - 305. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-160. + 277. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + 339.T + 1.03e5T^{2} \)
53 \( 1 - 349.T + 1.48e5T^{2} \)
59 \( 1 + (-129. + 224. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (325. - 563. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (447. + 775. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + (-370. + 642. i)T + (-1.78e5 - 3.09e5i)T^{2} \)
73 \( 1 + 820.T + 3.89e5T^{2} \)
79 \( 1 + 199.T + 4.93e5T^{2} \)
83 \( 1 + 541.T + 5.71e5T^{2} \)
89 \( 1 + (-190. - 329. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (715. - 1.23e3i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−12.90070168928051089986585853373, −11.99378065771217481049933286753, −10.46424437194428546757376296978, −9.246311554623732039243681514855, −7.67752714848159791653702992662, −7.03096098487148643184417859241, −6.26183906311374513443297861406, −5.14869089065033640316559204526, −4.06035172794972419340331495779, −1.46605091526454200811447183631, 1.57840848239689795647703803351, 2.73534072107228434748806407223, 4.31048790039981933213494462714, 5.37319403335498240696706126419, 5.96965657225511620656966945027, 8.560154871418416726786170043950, 9.614483457365576357085961989426, 10.37723171382327782829361658720, 11.29259540171130902603039579382, 11.89835765059421816130930005986

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