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Properties

Label 2-13e2-13.12-c7-0-7
Degree $2$
Conductor $169$
Sign $0.246 + 0.969i$
Analytic cond. $52.7930$
Root an. cond. $7.26588$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 19.6i·2-s − 72.4·3-s − 257.·4-s − 413. i·5-s − 1.42e3i·6-s + 1.00e3i·7-s − 2.53e3i·8-s + 3.06e3·9-s + 8.11e3·10-s + 3.48e3i·11-s + 1.86e4·12-s − 1.97e4·14-s + 2.99e4i·15-s + 1.68e4·16-s − 6.15e3·17-s + 6.01e4i·18-s + ⋯
L(s)  = 1  + 1.73i·2-s − 1.54·3-s − 2.00·4-s − 1.48i·5-s − 2.68i·6-s + 1.10i·7-s − 1.74i·8-s + 1.40·9-s + 2.56·10-s + 0.788i·11-s + 3.11·12-s − 1.92·14-s + 2.29i·15-s + 1.02·16-s − 0.303·17-s + 2.43i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.246 + 0.969i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.246 + 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(169\)    =    \(13^{2}\)
Sign: $0.246 + 0.969i$
Analytic conductor: \(52.7930\)
Root analytic conductor: \(7.26588\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{169} (168, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 169,\ (\ :7/2),\ 0.246 + 0.969i)\)

Particular Values

\(L(4)\) \(\approx\) \(0.3473984146\)
\(L(\frac12)\) \(\approx\) \(0.3473984146\)
\(L(\frac{9}{2})\) 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 \( 1 - 19.6iT - 128T^{2} \)
3 \( 1 + 72.4T + 2.18e3T^{2} \)
5 \( 1 + 413. iT - 7.81e4T^{2} \)
7 \( 1 - 1.00e3iT - 8.23e5T^{2} \)
11 \( 1 - 3.48e3iT - 1.94e7T^{2} \)
17 \( 1 + 6.15e3T + 4.10e8T^{2} \)
19 \( 1 - 4.14e4iT - 8.93e8T^{2} \)
23 \( 1 - 6.31e4T + 3.40e9T^{2} \)
29 \( 1 - 7.51e4T + 1.72e10T^{2} \)
31 \( 1 - 1.33e5iT - 2.75e10T^{2} \)
37 \( 1 - 1.71e5iT - 9.49e10T^{2} \)
41 \( 1 - 7.47e5iT - 1.94e11T^{2} \)
43 \( 1 - 1.87e5T + 2.71e11T^{2} \)
47 \( 1 + 2.26e5iT - 5.06e11T^{2} \)
53 \( 1 + 1.34e6T + 1.17e12T^{2} \)
59 \( 1 + 2.50e6iT - 2.48e12T^{2} \)
61 \( 1 + 5.09e5T + 3.14e12T^{2} \)
67 \( 1 - 4.07e6iT - 6.06e12T^{2} \)
71 \( 1 - 2.56e5iT - 9.09e12T^{2} \)
73 \( 1 - 5.38e6iT - 1.10e13T^{2} \)
79 \( 1 - 5.18e6T + 1.92e13T^{2} \)
83 \( 1 + 5.32e6iT - 2.71e13T^{2} \)
89 \( 1 + 8.69e6iT - 4.42e13T^{2} \)
97 \( 1 + 3.76e6iT - 8.07e13T^{2} \)
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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

−12.51093962649296178477959669427, −11.58321521563468076732969835301, −9.906040506655325032015727267893, −8.917579321960762068162349682698, −8.070471818271077788097263058812, −6.71842351885329967917528104719, −5.85375815146812132258733447973, −5.09738863687875403514159850001, −4.58678126790499134535816905482, −1.25200992463072740043971154339, 0.16546893688342005974362134973, 0.911361078239831954019097088255, 2.61787490388716319399267259029, 3.73988974833626095740925269069, 4.88079537785908184590519504944, 6.36028528516263733370021909967, 7.26314566552125032027356397042, 9.268023804459677606836722670634, 10.55488992558382464989946519330, 10.82537748264663546341752192524

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