Properties

Label 2-13e2-13.3-c3-0-0
Degree $2$
Conductor $169$
Sign $0.477 - 0.878i$
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  + (−0.780 − 1.35i)2-s + (−4.34 − 7.52i)3-s + (2.78 − 4.81i)4-s + 3.56·5-s + (−6.78 + 11.7i)6-s + (−13.5 + 23.5i)7-s − 21.1·8-s + (−24.2 + 41.9i)9-s + (−2.78 − 4.81i)10-s + (7.63 + 13.2i)11-s − 48.3·12-s + 42.4·14-s + (−15.4 − 26.7i)15-s + (−5.71 − 9.89i)16-s + (−22.2 + 38.5i)17-s + 75.6·18-s + ⋯
L(s)  = 1  + (−0.276 − 0.478i)2-s + (−0.835 − 1.44i)3-s + (0.347 − 0.602i)4-s + 0.318·5-s + (−0.461 + 0.799i)6-s + (−0.733 + 1.27i)7-s − 0.935·8-s + (−0.896 + 1.55i)9-s + (−0.0879 − 0.152i)10-s + (0.209 + 0.362i)11-s − 1.16·12-s + 0.810·14-s + (−0.266 − 0.461i)15-s + (−0.0892 − 0.154i)16-s + (−0.317 + 0.550i)17-s + 0.990·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.477 - 0.878i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.477 - 0.878i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(169\)    =    \(13^{2}\)
Sign: $0.477 - 0.878i$
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: \((2,\ 169,\ (\ :3/2),\ 0.477 - 0.878i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.0918549 + 0.0546148i\)
\(L(\frac12)\) \(\approx\) \(0.0918549 + 0.0546148i\)
\(L(\frac{5}{2})\) 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 \( 1 + (0.780 + 1.35i)T + (-4 + 6.92i)T^{2} \)
3 \( 1 + (4.34 + 7.52i)T + (-13.5 + 23.3i)T^{2} \)
5 \( 1 - 3.56T + 125T^{2} \)
7 \( 1 + (13.5 - 23.5i)T + (-171.5 - 297. i)T^{2} \)
11 \( 1 + (-7.63 - 13.2i)T + (-665.5 + 1.15e3i)T^{2} \)
17 \( 1 + (22.2 - 38.5i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-11.9 + 20.7i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (61.3 + 106. i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-109. - 190. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + 27.0T + 2.97e4T^{2} \)
37 \( 1 + (-47.0 - 81.5i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (80.1 + 138. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-75.6 + 131. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + 466.T + 1.03e5T^{2} \)
53 \( 1 + 120.T + 1.48e5T^{2} \)
59 \( 1 + (219. - 380. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-68.6 + 118. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-256. - 443. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + (-205. + 355. i)T + (-1.78e5 - 3.09e5i)T^{2} \)
73 \( 1 - 308.T + 3.89e5T^{2} \)
79 \( 1 + 586.T + 4.93e5T^{2} \)
83 \( 1 + 1.35e3T + 5.71e5T^{2} \)
89 \( 1 + (-219. - 380. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (755. - 1.30e3i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−12.29514119120070759456290854696, −11.75781572549167738840235861322, −10.67935371420945163867525584563, −9.584107117301959554132767748308, −8.460314949510064302249630975170, −6.83234721273674783448391046187, −6.22630619640177156132516453931, −5.40891434115284325877930497151, −2.57945313602027057362813716387, −1.59441261236114764025577277313, 0.05705526035593927403444570681, 3.35580831474923524967147856710, 4.26560926106881548762164216511, 5.81336225753821492746555505449, 6.68602040452028694379964110969, 7.959670871159863295939927607386, 9.499330563596735248370595637320, 9.935864277078041338210268166744, 11.12173528391288311412086578704, 11.77433635153107783688387945396

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