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Properties

Label 2-13e2-13.9-c1-0-5
Degree $2$
Conductor $169$
Sign $0.978 - 0.207i$
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  + (−0.400 + 0.694i)2-s + (1.12 − 1.94i)3-s + (0.678 + 1.17i)4-s + 0.246·5-s + (0.900 + 1.56i)6-s + (1.17 + 2.04i)7-s − 2.69·8-s + (−1.02 − 1.77i)9-s + (−0.0990 + 0.171i)10-s + (2.12 − 3.67i)11-s + 3.04·12-s − 1.89·14-s + (0.277 − 0.480i)15-s + (−0.277 + 0.480i)16-s + (−1.07 − 1.86i)17-s + 1.64·18-s + ⋯
L(s)  = 1  + (−0.283 + 0.491i)2-s + (0.648 − 1.12i)3-s + (0.339 + 0.587i)4-s + 0.110·5-s + (0.367 + 0.637i)6-s + (0.445 + 0.771i)7-s − 0.951·8-s + (−0.341 − 0.591i)9-s + (−0.0313 + 0.0542i)10-s + (0.640 − 1.10i)11-s + 0.880·12-s − 0.505·14-s + (0.0716 − 0.124i)15-s + (−0.0693 + 0.120i)16-s + (−0.261 − 0.453i)17-s + 0.387·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.978 - 0.207i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.978 - 0.207i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(169\)    =    \(13^{2}\)
Sign: $0.978 - 0.207i$
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: \((2,\ 169,\ (\ :1/2),\ 0.978 - 0.207i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.32284 + 0.138673i\)
\(L(\frac12)\) \(\approx\) \(1.32284 + 0.138673i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.400 - 0.694i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (-1.12 + 1.94i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 - 0.246T + 5T^{2} \)
7 \( 1 + (-1.17 - 2.04i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.12 + 3.67i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (1.07 + 1.86i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.0440 - 0.0763i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.746 - 1.29i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.31 - 4.01i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 6.63T + 31T^{2} \)
37 \( 1 + (2.84 - 4.92i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-5.79 + 10.0i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.147 - 0.256i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 7.35T + 47T^{2} \)
53 \( 1 + 10.3T + 53T^{2} \)
59 \( 1 + (-3.39 - 5.87i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.73 + 3.00i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.83 - 6.65i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-4.33 - 7.50i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 6.73T + 73T^{2} \)
79 \( 1 - 9.97T + 79T^{2} \)
83 \( 1 - 1.60T + 83T^{2} \)
89 \( 1 + (-1.44 + 2.49i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-4.02 - 6.97i)T + (-48.5 + 84.0i)T^{2} \)
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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

−12.82128398735163484804635421268, −11.95763065702170905163680686384, −11.17412388686513396905697310258, −9.183746466242397319474105672378, −8.530444674402549552489123486370, −7.69465836312251477138952191050, −6.76071270683026384646641741446, −5.68057615919087888106374482128, −3.38547592000766681510887012689, −2.03022052234198399630783108219, 1.92326429849570522147302022287, 3.70093089774374010984788368137, 4.74425255097343803165234389124, 6.35887132005009761752043214444, 7.74972249820632445027461222683, 9.235577118862531665770926662706, 9.713932374186595133810085011337, 10.59155196244628245608927553637, 11.37380838180154788326437431905, 12.61874394606007137537726973140

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