Properties

Label 2-13e2-13.9-c1-0-0
Degree $2$
Conductor $169$
Sign $-0.664 + 0.746i$
Analytic cond. $1.34947$
Root an. cond. $1.16166$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.866 + 1.5i)2-s + (−1 + 1.73i)3-s + (−0.5 − 0.866i)4-s − 1.73·5-s + (−1.73 − 3i)6-s − 1.73·8-s + (−0.499 − 0.866i)9-s + (1.49 − 2.59i)10-s + 2·12-s + (1.73 − 2.99i)15-s + (2.49 − 4.33i)16-s + (−1.5 − 2.59i)17-s + 1.73·18-s + (1.73 + 3i)19-s + (0.866 + 1.50i)20-s + ⋯
L(s)  = 1  + (−0.612 + 1.06i)2-s + (−0.577 + 0.999i)3-s + (−0.250 − 0.433i)4-s − 0.774·5-s + (−0.707 − 1.22i)6-s − 0.612·8-s + (−0.166 − 0.288i)9-s + (0.474 − 0.821i)10-s + 0.577·12-s + (0.447 − 0.774i)15-s + (0.624 − 1.08i)16-s + (−0.363 − 0.630i)17-s + 0.408·18-s + (0.397 + 0.688i)19-s + (0.193 + 0.335i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.664 + 0.746i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.664 + 0.746i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(169\)    =    \(13^{2}\)
Sign: $-0.664 + 0.746i$
Analytic conductor: \(1.34947\)
Root analytic conductor: \(1.16166\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{169} (22, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 169,\ (\ :1/2),\ -0.664 + 0.746i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.165213 - 0.368281i\)
\(L(\frac12)\) \(\approx\) \(0.165213 - 0.368281i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.866 - 1.5i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (1 - 1.73i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + 1.73T + 5T^{2} \)
7 \( 1 + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (1.5 + 2.59i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.73 - 3i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (3 - 5.19i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.5 - 2.59i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 3.46T + 31T^{2} \)
37 \( 1 + (4.33 - 7.5i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (2.59 - 4.5i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4 - 6.92i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 3.46T + 47T^{2} \)
53 \( 1 + 3T + 53T^{2} \)
59 \( 1 + (-3.46 - 6i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.73 + 3i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (1.73 + 3i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 1.73T + 73T^{2} \)
79 \( 1 - 4T + 79T^{2} \)
83 \( 1 - 13.8T + 83T^{2} \)
89 \( 1 + (-3.46 + 6i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (3.46 + 6i)T + (-48.5 + 84.0i)T^{2} \)
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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

−13.63354665267225313901414865356, −12.02330629613270342084878420756, −11.44949318530580681882544735116, −10.16326168720162575464679537427, −9.349665201872342313588955562620, −8.138632461951013330312426057722, −7.34049854202209080245702459206, −6.03641800405607795597681235072, −4.92344190111112481042795812599, −3.55982381203645878803030763207, 0.48155172633364216074792975011, 2.16476351842961474470588166149, 3.91237810568063505552473766069, 5.86009954120082912247415771514, 6.96304453987406784930278395185, 8.108558979652241302615090395837, 9.199571952202992873146175998705, 10.46892661763544810914590236874, 11.26444772344684466916008131603, 12.11720908568289323627167364369

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