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Properties

Label 2-13e2-13.10-c3-0-18
Degree $2$
Conductor $169$
Sign $0.997 + 0.0771i$
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  + (−2.21 + 1.28i)2-s + (1.84 + 3.19i)3-s + (−0.719 + 1.24i)4-s − 0.561i·5-s + (−8.17 − 4.71i)6-s + (−15.7 − 9.08i)7-s − 24.1i·8-s + (6.71 − 11.6i)9-s + (0.719 + 1.24i)10-s + (56.0 − 32.3i)11-s − 5.30·12-s + 46.5·14-s + (1.79 − 1.03i)15-s + (25.2 + 43.6i)16-s + (−12.7 + 22.1i)17-s + 34.3i·18-s + ⋯
L(s)  = 1  + (−0.784 + 0.452i)2-s + (0.354 + 0.614i)3-s + (−0.0899 + 0.155i)4-s − 0.0502i·5-s + (−0.556 − 0.321i)6-s + (−0.849 − 0.490i)7-s − 1.06i·8-s + (0.248 − 0.430i)9-s + (0.0227 + 0.0393i)10-s + (1.53 − 0.887i)11-s − 0.127·12-s + 0.888·14-s + (0.0308 − 0.0178i)15-s + (0.393 + 0.682i)16-s + (−0.182 + 0.315i)17-s + 0.450i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0771i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.997 + 0.0771i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(169\)    =    \(13^{2}\)
Sign: $0.997 + 0.0771i$
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: \((2,\ 169,\ (\ :3/2),\ 0.997 + 0.0771i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.01848 - 0.0393469i\)
\(L(\frac12)\) \(\approx\) \(1.01848 - 0.0393469i\)
\(L(\frac{5}{2})\) 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 \( 1 + (2.21 - 1.28i)T + (4 - 6.92i)T^{2} \)
3 \( 1 + (-1.84 - 3.19i)T + (-13.5 + 23.3i)T^{2} \)
5 \( 1 + 0.561iT - 125T^{2} \)
7 \( 1 + (15.7 + 9.08i)T + (171.5 + 297. i)T^{2} \)
11 \( 1 + (-56.0 + 32.3i)T + (665.5 - 1.15e3i)T^{2} \)
17 \( 1 + (12.7 - 22.1i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (93.5 + 53.9i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-36.6 - 63.4i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (87.9 + 152. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 - 113. iT - 2.97e4T^{2} \)
37 \( 1 + (-99.4 + 57.4i)T + (2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (-60.3 + 34.8i)T + (3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-219. + 379. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + 31.9iT - 1.03e5T^{2} \)
53 \( 1 - 2.84T + 1.48e5T^{2} \)
59 \( 1 + (62.0 + 35.8i)T + (1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-460. + 797. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-384. + 222. i)T + (1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + (469. + 270. i)T + (1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 - 764. iT - 3.89e5T^{2} \)
79 \( 1 + 421.T + 4.93e5T^{2} \)
83 \( 1 + 603. iT - 5.71e5T^{2} \)
89 \( 1 + (1.00e3 - 579. i)T + (3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (-505. - 291. i)T + (4.56e5 + 7.90e5i)T^{2} \)
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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

−12.42865742996374326282293298674, −11.03477024577893319118115507600, −9.895334731578507496629616059301, −9.122571957976248000280007248481, −8.589297500221279300426171037207, −7.01366615038148133529538597437, −6.36390549188519336866619975163, −4.14691272862163332164287113055, −3.46279201024160671435768254264, −0.67352219630030157809047347692, 1.34090027971695999753591062408, 2.54061455923627365166880529703, 4.48878104232202627639363190747, 6.18405356752296220082069813466, 7.19187933521122385628328991194, 8.552565392492754729469206989002, 9.265937067691313977717453731993, 10.14052173054430374042360056922, 11.21455106870432829627896734304, 12.39879717275714185877617643246

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