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Properties

Label 2-13e2-13.12-c3-0-22
Degree $2$
Conductor $169$
Sign $0.832 - 0.554i$
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  + 1.56i·2-s + 8.68·3-s + 5.56·4-s + 3.56i·5-s + 13.5i·6-s − 27.1i·7-s + 21.1i·8-s + 48.4·9-s − 5.56·10-s + 15.2i·11-s + 48.3·12-s + 42.4·14-s + 30.9i·15-s + 11.4·16-s − 44.5·17-s + 75.6i·18-s + ⋯
L(s)  = 1  + 0.552i·2-s + 1.67·3-s + 0.695·4-s + 0.318i·5-s + 0.922i·6-s − 1.46i·7-s + 0.935i·8-s + 1.79·9-s − 0.175·10-s + 0.418i·11-s + 1.16·12-s + 0.810·14-s + 0.532i·15-s + 0.178·16-s − 0.635·17-s + 0.990i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 - 0.554i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.832 - 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(169\)    =    \(13^{2}\)
Sign: $0.832 - 0.554i$
Analytic conductor: \(9.97132\)
Root analytic conductor: \(3.15774\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{169} (168, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 169,\ (\ :3/2),\ 0.832 - 0.554i)\)

Particular Values

\(L(2)\) \(\approx\) \(3.25541 + 0.985659i\)
\(L(\frac12)\) \(\approx\) \(3.25541 + 0.985659i\)
\(L(\frac{5}{2})\) 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 \( 1 - 1.56iT - 8T^{2} \)
3 \( 1 - 8.68T + 27T^{2} \)
5 \( 1 - 3.56iT - 125T^{2} \)
7 \( 1 + 27.1iT - 343T^{2} \)
11 \( 1 - 15.2iT - 1.33e3T^{2} \)
17 \( 1 + 44.5T + 4.91e3T^{2} \)
19 \( 1 + 23.9iT - 6.85e3T^{2} \)
23 \( 1 + 122.T + 1.21e4T^{2} \)
29 \( 1 + 219.T + 2.43e4T^{2} \)
31 \( 1 + 27.0iT - 2.97e4T^{2} \)
37 \( 1 - 94.1iT - 5.06e4T^{2} \)
41 \( 1 - 160. iT - 6.89e4T^{2} \)
43 \( 1 - 151.T + 7.95e4T^{2} \)
47 \( 1 - 466. iT - 1.03e5T^{2} \)
53 \( 1 + 120.T + 1.48e5T^{2} \)
59 \( 1 + 439. iT - 2.05e5T^{2} \)
61 \( 1 + 137.T + 2.26e5T^{2} \)
67 \( 1 + 512. iT - 3.00e5T^{2} \)
71 \( 1 + 410. iT - 3.57e5T^{2} \)
73 \( 1 + 308. iT - 3.89e5T^{2} \)
79 \( 1 + 586.T + 4.93e5T^{2} \)
83 \( 1 + 1.35e3iT - 5.71e5T^{2} \)
89 \( 1 - 439. iT - 7.04e5T^{2} \)
97 \( 1 - 1.51e3iT - 9.12e5T^{2} \)
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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

−12.79569519823317894836571779780, −11.19002429015146604790225504275, −10.29842150962359852082001974511, −9.222280383400328293896346714195, −7.925493051450372589442712002388, −7.43800871816415084487717507456, −6.50969599373457338364005981740, −4.39773827778505177266338267525, −3.17593117219127553110401518977, −1.86779802206111932031272950288, 1.88938369549942609959082339512, 2.68168933576581225898558715539, 3.84192285559113990002698616417, 5.78163970154381390325934767972, 7.23537529302004205765888669876, 8.440084321127611469370510437575, 9.035308703171759143244018235721, 10.02734623928198215316001190805, 11.33757229300034939139097567910, 12.36870417737929655037426606188

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