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Properties

Label 2-13e2-13.4-c3-0-29
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  + (1.92 + 1.11i)2-s + (4.87 − 8.44i)3-s + (−1.51 − 2.62i)4-s + 8.20i·5-s + (18.8 − 10.8i)6-s + (7.23 − 4.17i)7-s − 24.5i·8-s + (−34.0 − 58.9i)9-s + (−9.14 + 15.8i)10-s + (8.39 + 4.84i)11-s − 29.5·12-s + 18.6·14-s + (69.2 + 40.0i)15-s + (15.2 − 26.4i)16-s + (22.3 + 38.6i)17-s − 151. i·18-s + ⋯
L(s)  = 1  + (0.682 + 0.393i)2-s + (0.938 − 1.62i)3-s + (−0.189 − 0.328i)4-s + 0.734i·5-s + (1.27 − 0.738i)6-s + (0.390 − 0.225i)7-s − 1.08i·8-s + (−1.25 − 2.18i)9-s + (−0.289 + 0.500i)10-s + (0.230 + 0.132i)11-s − 0.712·12-s + 0.355·14-s + (1.19 + 0.688i)15-s + (0.238 − 0.412i)16-s + (0.318 + 0.551i)17-s − 1.98i·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

\(L(2)\) \(\approx\) \(2.10741 - 2.10651i\)
\(L(\frac12)\) \(\approx\) \(2.10741 - 2.10651i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-1.92 - 1.11i)T + (4 + 6.92i)T^{2} \)
3 \( 1 + (-4.87 + 8.44i)T + (-13.5 - 23.3i)T^{2} \)
5 \( 1 - 8.20iT - 125T^{2} \)
7 \( 1 + (-7.23 + 4.17i)T + (171.5 - 297. i)T^{2} \)
11 \( 1 + (-8.39 - 4.84i)T + (665.5 + 1.15e3i)T^{2} \)
17 \( 1 + (-22.3 - 38.6i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (75.9 - 43.8i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-53.5 + 92.7i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-7.02 + 12.1i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + 171. iT - 2.97e4T^{2} \)
37 \( 1 + (-358. - 206. i)T + (2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (-223. - 129. i)T + (3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-30.5 - 52.8i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + 68.7iT - 1.03e5T^{2} \)
53 \( 1 - 328.T + 1.48e5T^{2} \)
59 \( 1 + (-127. + 73.5i)T + (1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-48.9 - 84.7i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-586. - 338. i)T + (1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + (681. - 393. i)T + (1.78e5 - 3.09e5i)T^{2} \)
73 \( 1 - 997. iT - 3.89e5T^{2} \)
79 \( 1 - 383.T + 4.93e5T^{2} \)
83 \( 1 + 519. iT - 5.71e5T^{2} \)
89 \( 1 + (591. + 341. i)T + (3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (-300. + 173. i)T + (4.56e5 - 7.90e5i)T^{2} \)
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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

−12.67909912471033597891605967114, −11.33208657837561345237143602165, −9.975971211143836067677403973724, −8.658756551388154103989491725682, −7.66626419771649424143863185493, −6.69348328457344812051909784226, −6.04541387178706561014042217345, −4.11446860456000053494806583252, −2.63630110255261905305745129860, −1.11119586646441636286283993731, 2.53354468920178972334872807734, 3.69860464194582576018147025181, 4.64631626248557639926151554368, 5.31737656498364220826654026213, 7.85764537713739568720450131801, 8.836107474423275530251576286914, 9.258645351464239754625229694569, 10.67223087555292345369374565447, 11.49394249589106958305078489302, 12.71603285792602337387711390142

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