Properties

Label 2-13e2-13.3-c3-0-1
Degree $2$
Conductor $169$
Sign $0.859 + 0.511i$
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  + (−2.28 − 3.95i)2-s + (−4.34 − 7.52i)3-s + (−6.40 + 11.0i)4-s − 2.80·5-s + (−19.8 + 34.3i)6-s + (4.78 − 8.28i)7-s + 21.9·8-s + (−24.2 + 41.9i)9-s + (6.40 + 11.0i)10-s + (19.7 + 34.1i)11-s + 111.·12-s − 43.6·14-s + (12.1 + 21.1i)15-s + (1.21 + 2.09i)16-s + (−1.00 + 1.74i)17-s + 220.·18-s + ⋯
L(s)  = 1  + (−0.806 − 1.39i)2-s + (−0.835 − 1.44i)3-s + (−0.800 + 1.38i)4-s − 0.251·5-s + (−1.34 + 2.33i)6-s + (0.258 − 0.447i)7-s + 0.969·8-s + (−0.896 + 1.55i)9-s + (0.202 + 0.350i)10-s + (0.540 + 0.935i)11-s + 2.67·12-s − 0.832·14-s + (0.209 + 0.363i)15-s + (0.0189 + 0.0327i)16-s + (−0.0143 + 0.0248i)17-s + 2.89·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.859 + 0.511i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.859 + 0.511i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(169\)    =    \(13^{2}\)
Sign: $0.859 + 0.511i$
Analytic conductor: \(9.97132\)
Root analytic conductor: \(3.15774\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{169} (146, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 169,\ (\ :3/2),\ 0.859 + 0.511i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.245364 - 0.0674342i\)
\(L(\frac12)\) \(\approx\) \(0.245364 - 0.0674342i\)
\(L(\frac{5}{2})\) 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 \( 1 + (2.28 + 3.95i)T + (-4 + 6.92i)T^{2} \)
3 \( 1 + (4.34 + 7.52i)T + (-13.5 + 23.3i)T^{2} \)
5 \( 1 + 2.80T + 125T^{2} \)
7 \( 1 + (-4.78 + 8.28i)T + (-171.5 - 297. i)T^{2} \)
11 \( 1 + (-19.7 - 34.1i)T + (-665.5 + 1.15e3i)T^{2} \)
17 \( 1 + (1.00 - 1.74i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (30.0 - 52.1i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (2.23 + 3.87i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (70.3 + 121. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + 136.T + 2.97e4T^{2} \)
37 \( 1 + (92.8 + 160. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (-155. - 268. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (213. - 370. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 - 258.T + 1.03e5T^{2} \)
53 \( 1 - 612.T + 1.48e5T^{2} \)
59 \( 1 + (258. - 448. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-80.6 + 139. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (24.9 + 43.2i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + (-139. + 242. i)T + (-1.78e5 - 3.09e5i)T^{2} \)
73 \( 1 + 467.T + 3.89e5T^{2} \)
79 \( 1 - 37.5T + 4.93e5T^{2} \)
83 \( 1 - 76.1T + 5.71e5T^{2} \)
89 \( 1 + (-101. - 175. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (587. - 1.01e3i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−12.04532473168950856793784912943, −11.41413633002123394857428154784, −10.53548653588220940773710075672, −9.425580948660905362410350534125, −8.048316816494683575238330832653, −7.28892050543763734369070124205, −5.98475556273315445405995093610, −4.09739405464342600629541873438, −2.15282321316503580757975705579, −1.19208623892088374915954199560, 0.19575420871275689320397069220, 3.80016445874416820957361758546, 5.23286414373089117967925525801, 5.86725205945025099084821119317, 7.07101893973602467579854592384, 8.587119821431368684782353181580, 9.097490518064933571677797586930, 10.17829377649509173197405104720, 11.13361578505531048641298812526, 12.00979767309549647355551975653

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