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Properties

Label 2-13e2-13.10-c3-0-25
Degree $2$
Conductor $169$
Sign $-0.542 + 0.839i$
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  + (−0.337 + 0.195i)2-s + (−1.80 − 3.12i)3-s + (−3.92 + 6.79i)4-s − 7.52i·5-s + (1.21 + 0.704i)6-s + (16.9 + 9.77i)7-s − 6.18i·8-s + (6.98 − 12.0i)9-s + (1.46 + 2.54i)10-s + (−39.6 + 22.9i)11-s + 28.3·12-s − 7.62·14-s + (−23.5 + 13.5i)15-s + (−30.1 − 52.2i)16-s + (43.2 − 74.9i)17-s + 5.45i·18-s + ⋯
L(s)  = 1  + (−0.119 + 0.0689i)2-s + (−0.347 − 0.601i)3-s + (−0.490 + 0.849i)4-s − 0.672i·5-s + (0.0829 + 0.0479i)6-s + (0.913 + 0.527i)7-s − 0.273i·8-s + (0.258 − 0.448i)9-s + (0.0463 + 0.0803i)10-s + (−1.08 + 0.628i)11-s + 0.681·12-s − 0.145·14-s + (−0.404 + 0.233i)15-s + (−0.471 − 0.816i)16-s + (0.617 − 1.06i)17-s + 0.0713i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.542 + 0.839i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.542 + 0.839i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(169\)    =    \(13^{2}\)
Sign: $-0.542 + 0.839i$
Analytic conductor: \(9.97132\)
Root analytic conductor: \(3.15774\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{169} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 169,\ (\ :3/2),\ -0.542 + 0.839i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.361767 - 0.664660i\)
\(L(\frac12)\) \(\approx\) \(0.361767 - 0.664660i\)
\(L(\frac{5}{2})\) 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 \( 1 + (0.337 - 0.195i)T + (4 - 6.92i)T^{2} \)
3 \( 1 + (1.80 + 3.12i)T + (-13.5 + 23.3i)T^{2} \)
5 \( 1 + 7.52iT - 125T^{2} \)
7 \( 1 + (-16.9 - 9.77i)T + (171.5 + 297. i)T^{2} \)
11 \( 1 + (39.6 - 22.9i)T + (665.5 - 1.15e3i)T^{2} \)
17 \( 1 + (-43.2 + 74.9i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (128. + 74.3i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (45.7 + 79.2i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (129. + 224. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 - 31.2iT - 2.97e4T^{2} \)
37 \( 1 + (128. - 74.2i)T + (2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (-83.0 + 47.9i)T + (3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (40.4 - 70.1i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + 94.3iT - 1.03e5T^{2} \)
53 \( 1 - 493.T + 1.48e5T^{2} \)
59 \( 1 + (498. + 287. i)T + (1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-20.1 + 34.8i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-520. + 300. i)T + (1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + (-449. - 259. i)T + (1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 - 1.05e3iT - 3.89e5T^{2} \)
79 \( 1 + 320.T + 4.93e5T^{2} \)
83 \( 1 + 32.4iT - 5.71e5T^{2} \)
89 \( 1 + (390. - 225. i)T + (3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (200. + 115. i)T + (4.56e5 + 7.90e5i)T^{2} \)
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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

−12.27452262516125985402069534627, −11.29297506949881729754214198793, −9.767781566992784560056888598051, −8.681889304421452392001716646245, −7.920512773172739306256083660294, −6.90293174734808909686123580436, −5.26678640619717076519800331839, −4.35833406010057749055873029225, −2.34517366001080145481899408429, −0.38063165813259356060708890348, 1.74103143499895697434102946225, 3.90659460999655164653623293737, 5.08532542493321233859148299284, 5.95919038231444150952032943535, 7.61035398665380304189684458156, 8.580771886414386151914964144909, 10.11499306115633554405700089582, 10.72015198077348382256431993544, 10.96259667767805441434163434039, 12.76655856764305732523117338894

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