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Properties

Label 2-13e2-13.3-c3-0-8
Degree $2$
Conductor $169$
Sign $-0.945 + 0.326i$
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  + (1.73 + 3i)2-s + (3.5 + 6.06i)3-s + (−2 + 3.46i)4-s − 13.8·5-s + (−12.1 + 21i)6-s + (−11.2 + 19.5i)7-s + 13.8·8-s + (−11 + 19.0i)9-s + (−23.9 − 41.5i)10-s + (−11.2 − 19.5i)11-s − 28.0·12-s − 78·14-s + (−48.4 − 84i)15-s + (39.9 + 69.2i)16-s + (13.5 − 23.3i)17-s − 76.2·18-s + ⋯
L(s)  = 1  + (0.612 + 1.06i)2-s + (0.673 + 1.16i)3-s + (−0.250 + 0.433i)4-s − 1.23·5-s + (−0.824 + 1.42i)6-s + (−0.607 + 1.05i)7-s + 0.612·8-s + (−0.407 + 0.705i)9-s + (−0.758 − 1.31i)10-s + (−0.308 − 0.534i)11-s − 0.673·12-s − 1.48·14-s + (−0.834 − 1.44i)15-s + (0.624 + 1.08i)16-s + (0.192 − 0.333i)17-s − 0.997·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.945 + 0.326i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.945 + 0.326i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(169\)    =    \(13^{2}\)
Sign: $-0.945 + 0.326i$
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: \((2,\ 169,\ (\ :3/2),\ -0.945 + 0.326i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.335163 - 1.99670i\)
\(L(\frac12)\) \(\approx\) \(0.335163 - 1.99670i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-1.73 - 3i)T + (-4 + 6.92i)T^{2} \)
3 \( 1 + (-3.5 - 6.06i)T + (-13.5 + 23.3i)T^{2} \)
5 \( 1 + 13.8T + 125T^{2} \)
7 \( 1 + (11.2 - 19.5i)T + (-171.5 - 297. i)T^{2} \)
11 \( 1 + (11.2 + 19.5i)T + (-665.5 + 1.15e3i)T^{2} \)
17 \( 1 + (-13.5 + 23.3i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (44.1 - 76.5i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-28.5 - 49.3i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-34.5 - 59.7i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 - 72.7T + 2.97e4T^{2} \)
37 \( 1 + (19.9 + 34.5i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (-196. - 340.5i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (42.5 - 73.6i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + 342.T + 1.03e5T^{2} \)
53 \( 1 - 426T + 1.48e5T^{2} \)
59 \( 1 + (-9.52 + 16.5i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-8.5 + 14.7i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (82.2 + 142.5i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + (-291. + 505.5i)T + (-1.78e5 - 3.09e5i)T^{2} \)
73 \( 1 - 1.00e3T + 3.89e5T^{2} \)
79 \( 1 + 1.24e3T + 4.93e5T^{2} \)
83 \( 1 - 426.T + 5.71e5T^{2} \)
89 \( 1 + (-153. - 265.5i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (-617. + 1.06e3i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−13.08077271646546253250017404290, −11.98962217258746655753109634959, −10.80894277250198753617968164107, −9.677857687529950795393269751824, −8.546305185147663937398107420501, −7.81685575874874361855510550476, −6.38352627037604110932431769694, −5.21215179297313945104997192734, −4.10360630340555716848383515633, −3.12004335684016734339341771507, 0.71893077149200097815950516107, 2.36986999567970864789516699977, 3.55471853931236530034756333824, 4.50593432421718688574708695636, 6.89820921126313904293861866205, 7.47224532272548667089043245728, 8.418041462657932276202687146163, 10.12276725304448429124908136488, 11.04556752892983942830455106656, 12.06569434718210944740294938151

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