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Properties

Label 2-13e2-13.11-c2-0-6
Degree $2$
Conductor $169$
Sign $-0.504 - 0.863i$
Analytic cond. $4.60491$
Root an. cond. $2.14590$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.515 + 1.92i)2-s + (0.299 + 0.519i)3-s + (0.0309 + 0.0178i)4-s + (5.65 + 5.65i)5-s + (−1.15 + 0.308i)6-s + (−0.526 − 1.96i)7-s + (−5.68 + 5.68i)8-s + (4.32 − 7.48i)9-s + (−13.7 + 7.95i)10-s + (14.0 + 3.76i)11-s + 0.0214i·12-s + 4.05·14-s + (−1.24 + 4.62i)15-s + (−7.92 − 13.7i)16-s + (−10.2 − 5.90i)17-s + (12.1 + 12.1i)18-s + ⋯
L(s)  = 1  + (−0.257 + 0.961i)2-s + (0.0999 + 0.173i)3-s + (0.00773 + 0.00446i)4-s + (1.13 + 1.13i)5-s + (−0.192 + 0.0514i)6-s + (−0.0752 − 0.280i)7-s + (−0.710 + 0.710i)8-s + (0.480 − 0.831i)9-s + (−1.37 + 0.795i)10-s + (1.27 + 0.342i)11-s + 0.00178i·12-s + 0.289·14-s + (−0.0826 + 0.308i)15-s + (−0.495 − 0.858i)16-s + (−0.601 − 0.347i)17-s + (0.675 + 0.675i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.504 - 0.863i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.504 - 0.863i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(169\)    =    \(13^{2}\)
Sign: $-0.504 - 0.863i$
Analytic conductor: \(4.60491\)
Root analytic conductor: \(2.14590\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{169} (89, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 169,\ (\ :1),\ -0.504 - 0.863i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.859149 + 1.49704i\)
\(L(\frac12)\) \(\approx\) \(0.859149 + 1.49704i\)
\(L(2)\) 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 \( 1 + (0.515 - 1.92i)T + (-3.46 - 2i)T^{2} \)
3 \( 1 + (-0.299 - 0.519i)T + (-4.5 + 7.79i)T^{2} \)
5 \( 1 + (-5.65 - 5.65i)T + 25iT^{2} \)
7 \( 1 + (0.526 + 1.96i)T + (-42.4 + 24.5i)T^{2} \)
11 \( 1 + (-14.0 - 3.76i)T + (104. + 60.5i)T^{2} \)
17 \( 1 + (10.2 + 5.90i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (23.4 - 6.28i)T + (312. - 180.5i)T^{2} \)
23 \( 1 + (-0.229 + 0.132i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (3.60 + 6.25i)T + (-420.5 + 728. i)T^{2} \)
31 \( 1 + (30.2 + 30.2i)T + 961iT^{2} \)
37 \( 1 + (-13.1 - 3.53i)T + (1.18e3 + 684.5i)T^{2} \)
41 \( 1 + (-9.00 + 33.6i)T + (-1.45e3 - 840.5i)T^{2} \)
43 \( 1 + (-35.0 - 20.2i)T + (924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (9.87 - 9.87i)T - 2.20e3iT^{2} \)
53 \( 1 - 77.1T + 2.80e3T^{2} \)
59 \( 1 + (9.32 + 34.8i)T + (-3.01e3 + 1.74e3i)T^{2} \)
61 \( 1 + (-15.4 + 26.7i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (16.1 - 60.4i)T + (-3.88e3 - 2.24e3i)T^{2} \)
71 \( 1 + (-8.34 + 2.23i)T + (4.36e3 - 2.52e3i)T^{2} \)
73 \( 1 + (-23.3 + 23.3i)T - 5.32e3iT^{2} \)
79 \( 1 + 49.8T + 6.24e3T^{2} \)
83 \( 1 + (60.7 + 60.7i)T + 6.88e3iT^{2} \)
89 \( 1 + (-85.6 - 22.9i)T + (6.85e3 + 3.96e3i)T^{2} \)
97 \( 1 + (32.6 - 8.75i)T + (8.14e3 - 4.70e3i)T^{2} \)
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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

−13.07103363967669989999479787732, −11.80724639404624003359222404248, −10.72590364964405311703242335589, −9.631693013635334598130668962226, −8.921276875808931986782856611631, −7.27393846156734682353893206190, −6.60661573277273417416939891517, −5.95526555596792784850381365019, −3.92691926232441695487836156234, −2.26053475659519885743776778064, 1.32284540291596326444219357799, 2.25872887536294057010202119139, 4.24998721295863761627849222269, 5.72106023101814798084479483230, 6.76924379550929555669288640842, 8.709553088459899940092482499455, 9.174014490529345492707623716000, 10.26370459682319378281878868735, 11.12409032943640179315681185309, 12.35498314887585470035288333655

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