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Properties

Label 2-13e2-13.4-c3-0-1
Degree $2$
Conductor $169$
Sign $-0.756 + 0.654i$
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  + (1.49 + 0.863i)2-s + (−3.44 + 5.97i)3-s + (−2.50 − 4.34i)4-s + 20.8i·5-s + (−10.3 + 5.95i)6-s + (−6.55 + 3.78i)7-s − 22.4i·8-s + (−10.2 − 17.7i)9-s + (−17.9 + 31.1i)10-s + (−3.81 − 2.20i)11-s + 34.5·12-s − 13.0·14-s + (−124. − 71.8i)15-s + (−0.637 + 1.10i)16-s + (−36.5 − 63.2i)17-s − 35.5i·18-s + ⋯
L(s)  = 1  + (0.528 + 0.305i)2-s + (−0.663 + 1.14i)3-s + (−0.313 − 0.542i)4-s + 1.86i·5-s + (−0.702 + 0.405i)6-s + (−0.353 + 0.204i)7-s − 0.993i·8-s + (−0.380 − 0.659i)9-s + (−0.568 + 0.985i)10-s + (−0.104 − 0.0603i)11-s + 0.831·12-s − 0.249·14-s + (−2.14 − 1.23i)15-s + (−0.00996 + 0.0172i)16-s + (−0.520 − 0.902i)17-s − 0.464i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.756 + 0.654i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.756 + 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(169\)    =    \(13^{2}\)
Sign: $-0.756 + 0.654i$
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: \((2,\ 169,\ (\ :3/2),\ -0.756 + 0.654i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.253377 - 0.680250i\)
\(L(\frac12)\) \(\approx\) \(0.253377 - 0.680250i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-1.49 - 0.863i)T + (4 + 6.92i)T^{2} \)
3 \( 1 + (3.44 - 5.97i)T + (-13.5 - 23.3i)T^{2} \)
5 \( 1 - 20.8iT - 125T^{2} \)
7 \( 1 + (6.55 - 3.78i)T + (171.5 - 297. i)T^{2} \)
11 \( 1 + (3.81 + 2.20i)T + (665.5 + 1.15e3i)T^{2} \)
17 \( 1 + (36.5 + 63.2i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-48.4 + 27.9i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (16.8 - 29.1i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (60.7 - 105. i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + 84.1iT - 2.97e4T^{2} \)
37 \( 1 + (-148. - 85.8i)T + (2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (81.0 + 46.7i)T + (3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-220. - 382. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + 272. iT - 1.03e5T^{2} \)
53 \( 1 + 480.T + 1.48e5T^{2} \)
59 \( 1 + (303. - 175. i)T + (1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-242. - 419. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-837. - 483. i)T + (1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + (348. - 201. i)T + (1.78e5 - 3.09e5i)T^{2} \)
73 \( 1 - 351. iT - 3.89e5T^{2} \)
79 \( 1 + 820.T + 4.93e5T^{2} \)
83 \( 1 - 192. iT - 5.71e5T^{2} \)
89 \( 1 + (704. + 406. i)T + (3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (-682. + 394. i)T + (4.56e5 - 7.90e5i)T^{2} \)
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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

−13.20692299722435706010191913491, −11.49526928333917260622449461639, −10.93073388419454125399185476991, −9.982254599367535324632672970407, −9.463270656728205850190278530429, −7.28788981597354986697728520123, −6.32703823911440202277156210146, −5.42972866024426256318376210618, −4.20060439658883039640197110221, −2.98340080641113582542993529322, 0.30416653407645712132401830983, 1.76608073934970630672729157869, 3.94915803601167038501963395752, 5.07015633048562138809442578575, 6.09547979900912661553864142556, 7.64564735831198955494869444821, 8.435031260860997835242284252274, 9.505050229142667559731956048879, 11.27092477750559191168294838912, 12.22013139596931927986447704188

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