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Properties

Label 2-13e2-13.12-c5-0-9
Degree $2$
Conductor $169$
Sign $-0.832 - 0.554i$
Analytic cond. $27.1048$
Root an. cond. $5.20623$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 10.9i·2-s − 15.7·3-s − 88.2·4-s − 51.6i·5-s − 173. i·6-s + 75.3i·7-s − 617. i·8-s + 6.60·9-s + 565.·10-s − 255. i·11-s + 1.39e3·12-s − 826.·14-s + 815. i·15-s + 3.94e3·16-s − 53.2·17-s + 72.4i·18-s + ⋯
L(s)  = 1  + 1.93i·2-s − 1.01·3-s − 2.75·4-s − 0.923i·5-s − 1.96i·6-s + 0.581i·7-s − 3.41i·8-s + 0.0271·9-s + 1.78·10-s − 0.637i·11-s + 2.79·12-s − 1.12·14-s + 0.935i·15-s + 3.85·16-s − 0.0446·17-s + 0.0527i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.832 - 0.554i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.832 - 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(169\)    =    \(13^{2}\)
Sign: $-0.832 - 0.554i$
Analytic conductor: \(27.1048\)
Root analytic conductor: \(5.20623\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{169} (168, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 169,\ (\ :5/2),\ -0.832 - 0.554i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.6466187598\)
\(L(\frac12)\) \(\approx\) \(0.6466187598\)
\(L(\frac{7}{2})\) 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 \( 1 - 10.9iT - 32T^{2} \)
3 \( 1 + 15.7T + 243T^{2} \)
5 \( 1 + 51.6iT - 3.12e3T^{2} \)
7 \( 1 - 75.3iT - 1.68e4T^{2} \)
11 \( 1 + 255. iT - 1.61e5T^{2} \)
17 \( 1 + 53.2T + 1.41e6T^{2} \)
19 \( 1 - 268. iT - 2.47e6T^{2} \)
23 \( 1 + 2.08e3T + 6.43e6T^{2} \)
29 \( 1 + 8.17e3T + 2.05e7T^{2} \)
31 \( 1 + 4.78e3iT - 2.86e7T^{2} \)
37 \( 1 - 6.65e3iT - 6.93e7T^{2} \)
41 \( 1 - 1.28e3iT - 1.15e8T^{2} \)
43 \( 1 + 3.80e3T + 1.47e8T^{2} \)
47 \( 1 + 3.77e3iT - 2.29e8T^{2} \)
53 \( 1 - 2.40e4T + 4.18e8T^{2} \)
59 \( 1 + 2.45e4iT - 7.14e8T^{2} \)
61 \( 1 - 3.51e4T + 8.44e8T^{2} \)
67 \( 1 - 1.50e4iT - 1.35e9T^{2} \)
71 \( 1 - 5.11e4iT - 1.80e9T^{2} \)
73 \( 1 - 6.30e4iT - 2.07e9T^{2} \)
79 \( 1 + 8.04e4T + 3.07e9T^{2} \)
83 \( 1 - 1.13e5iT - 3.93e9T^{2} \)
89 \( 1 - 6.95e4iT - 5.58e9T^{2} \)
97 \( 1 - 1.83e4iT - 8.58e9T^{2} \)
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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

−12.59059847596650502183070388339, −11.48236633733987063414624886062, −9.845785881645404414433187844358, −8.798067010275704141697303641447, −8.150300817339157952108573502432, −6.80907461794045305033506076685, −5.65528514764244330954536332366, −5.43833185562009643386344107340, −4.11792672879484767993094448712, −0.65566756307348732069666799273, 0.44013765132041552171750989143, 1.97668211115169813796600123598, 3.32650515316524424706104427831, 4.47690886385634012294365851830, 5.70444324549909213201909054849, 7.30299113107368097774964339076, 8.914985933346602987709773689450, 10.13164241213192171790617257775, 10.64711440963764768309611198232, 11.37168615854129567616204602193

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