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Properties

Label 2-13e2-13.12-c3-0-16
Degree $2$
Conductor $169$
Sign $-0.832 - 0.554i$
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  + 4.56i·2-s + 8.68·3-s − 12.8·4-s − 2.80i·5-s + 39.6i·6-s + 9.56i·7-s − 21.9i·8-s + 48.4·9-s + 12.8·10-s + 39.4i·11-s − 111.·12-s − 43.6·14-s − 24.3i·15-s − 2.42·16-s − 2.01·17-s + 220. i·18-s + ⋯
L(s)  = 1  + 1.61i·2-s + 1.67·3-s − 1.60·4-s − 0.251i·5-s + 2.69i·6-s + 0.516i·7-s − 0.969i·8-s + 1.79·9-s + 0.405·10-s + 1.08i·11-s − 2.67·12-s − 0.832·14-s − 0.419i·15-s − 0.0378·16-s − 0.0287·17-s + 2.89i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.832 - 0.554i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.832 - 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(169\)    =    \(13^{2}\)
Sign: $-0.832 - 0.554i$
Analytic conductor: \(9.97132\)
Root analytic conductor: \(3.15774\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{169} (168, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 169,\ (\ :3/2),\ -0.832 - 0.554i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.782577 + 2.58467i\)
\(L(\frac12)\) \(\approx\) \(0.782577 + 2.58467i\)
\(L(\frac{5}{2})\) 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 \( 1 - 4.56iT - 8T^{2} \)
3 \( 1 - 8.68T + 27T^{2} \)
5 \( 1 + 2.80iT - 125T^{2} \)
7 \( 1 - 9.56iT - 343T^{2} \)
11 \( 1 - 39.4iT - 1.33e3T^{2} \)
17 \( 1 + 2.01T + 4.91e3T^{2} \)
19 \( 1 - 60.1iT - 6.85e3T^{2} \)
23 \( 1 + 4.46T + 1.21e4T^{2} \)
29 \( 1 - 140.T + 2.43e4T^{2} \)
31 \( 1 + 136. iT - 2.97e4T^{2} \)
37 \( 1 + 185. iT - 5.06e4T^{2} \)
41 \( 1 + 310. iT - 6.89e4T^{2} \)
43 \( 1 + 427.T + 7.95e4T^{2} \)
47 \( 1 + 258. iT - 1.03e5T^{2} \)
53 \( 1 - 612.T + 1.48e5T^{2} \)
59 \( 1 + 517. iT - 2.05e5T^{2} \)
61 \( 1 + 161.T + 2.26e5T^{2} \)
67 \( 1 - 49.8iT - 3.00e5T^{2} \)
71 \( 1 + 279. iT - 3.57e5T^{2} \)
73 \( 1 - 467. iT - 3.89e5T^{2} \)
79 \( 1 - 37.5T + 4.93e5T^{2} \)
83 \( 1 - 76.1iT - 5.71e5T^{2} \)
89 \( 1 - 202. iT - 7.04e5T^{2} \)
97 \( 1 - 1.17e3iT - 9.12e5T^{2} \)
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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

−13.20939173019579476727431356775, −12.21451829364829241199856697944, −10.09290376998499119091692084964, −9.109873485649937062865630222199, −8.481263090631912848501989074009, −7.63247033710245601900340088075, −6.74878191153807085687488048469, −5.23193504129251305288830246955, −3.99540257240435191514380096063, −2.24598204701257982905979197878, 1.16928263155727204969620560755, 2.72962649891808148369347214836, 3.31982572340343047053478756121, 4.52076312914273031231803941619, 6.90367541249636820555335747385, 8.333306117364102748965948730040, 8.974202582625530949459316984411, 10.03411706426754121108194766680, 10.76943041222701393129096120205, 11.86284048019958281353605904986

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