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Properties

Label 2-13e2-13.4-c3-0-33
Degree $2$
Conductor $169$
Sign $-0.743 - 0.669i$
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  + (−1.92 − 1.11i)2-s + (4.87 − 8.44i)3-s + (−1.51 − 2.62i)4-s − 8.20i·5-s + (−18.8 + 10.8i)6-s + (−7.23 + 4.17i)7-s + 24.5i·8-s + (−34.0 − 58.9i)9-s + (−9.14 + 15.8i)10-s + (−8.39 − 4.84i)11-s − 29.5·12-s + 18.6·14-s + (−69.2 − 40.0i)15-s + (15.2 − 26.4i)16-s + (22.3 + 38.6i)17-s + 151. i·18-s + ⋯
L(s)  = 1  + (−0.682 − 0.393i)2-s + (0.938 − 1.62i)3-s + (−0.189 − 0.328i)4-s − 0.734i·5-s + (−1.27 + 0.738i)6-s + (−0.390 + 0.225i)7-s + 1.08i·8-s + (−1.25 − 2.18i)9-s + (−0.289 + 0.500i)10-s + (−0.230 − 0.132i)11-s − 0.712·12-s + 0.355·14-s + (−1.19 − 0.688i)15-s + (0.238 − 0.412i)16-s + (0.318 + 0.551i)17-s + 1.98i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.743 - 0.669i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.743 - 0.669i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(169\)    =    \(13^{2}\)
Sign: $-0.743 - 0.669i$
Analytic conductor: \(9.97132\)
Root analytic conductor: \(3.15774\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{169} (147, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 169,\ (\ :3/2),\ -0.743 - 0.669i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.398316 + 1.03782i\)
\(L(\frac12)\) \(\approx\) \(0.398316 + 1.03782i\)
\(L(\frac{5}{2})\) 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 \( 1 + (1.92 + 1.11i)T + (4 + 6.92i)T^{2} \)
3 \( 1 + (-4.87 + 8.44i)T + (-13.5 - 23.3i)T^{2} \)
5 \( 1 + 8.20iT - 125T^{2} \)
7 \( 1 + (7.23 - 4.17i)T + (171.5 - 297. i)T^{2} \)
11 \( 1 + (8.39 + 4.84i)T + (665.5 + 1.15e3i)T^{2} \)
17 \( 1 + (-22.3 - 38.6i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-75.9 + 43.8i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-53.5 + 92.7i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-7.02 + 12.1i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 - 171. iT - 2.97e4T^{2} \)
37 \( 1 + (358. + 206. i)T + (2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (223. + 129. i)T + (3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-30.5 - 52.8i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 - 68.7iT - 1.03e5T^{2} \)
53 \( 1 - 328.T + 1.48e5T^{2} \)
59 \( 1 + (127. - 73.5i)T + (1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-48.9 - 84.7i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (586. + 338. i)T + (1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + (-681. + 393. i)T + (1.78e5 - 3.09e5i)T^{2} \)
73 \( 1 + 997. iT - 3.89e5T^{2} \)
79 \( 1 - 383.T + 4.93e5T^{2} \)
83 \( 1 - 519. iT - 5.71e5T^{2} \)
89 \( 1 + (-591. - 341. i)T + (3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (300. - 173. i)T + (4.56e5 - 7.90e5i)T^{2} \)
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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

−12.03668240300631559757213165364, −10.60893172529358226006899470928, −9.171126417842759317986225845115, −8.773851036662149296392572936967, −7.83247793554020860065472878219, −6.64030127921148149727458993410, −5.30747743641415659223963185878, −3.01486576979165243726515133816, −1.68055323097425427983192930272, −0.59087934976777855668079618945, 3.03588410154132952530974961313, 3.72270692558337205925501749681, 5.14995321923509971448972489519, 7.05185911879013733867920338110, 8.025658798732513601756572441873, 9.010117765868191045618728818169, 9.861735322065931871755942506843, 10.31203223273505898912719880327, 11.62413618263080944088210827897, 13.29771444550781230994308777860

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