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Properties

Label 2-13e2-13.9-c3-0-19
Degree $2$
Conductor $169$
Sign $0.872 + 0.488i$
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  + (−1.5 + 2.59i)2-s + (0.5 − 0.866i)3-s + (−0.5 − 0.866i)4-s − 9·5-s + (1.5 + 2.59i)6-s + (−7.5 − 12.9i)7-s − 21·8-s + (13 + 22.5i)9-s + (13.5 − 23.3i)10-s + (24 − 41.5i)11-s − 1.00·12-s + 45·14-s + (−4.5 + 7.79i)15-s + (35.5 − 61.4i)16-s + (−22.5 − 38.9i)17-s − 78·18-s + ⋯
L(s)  = 1  + (−0.530 + 0.918i)2-s + (0.0962 − 0.166i)3-s + (−0.0625 − 0.108i)4-s − 0.804·5-s + (0.102 + 0.176i)6-s + (−0.404 − 0.701i)7-s − 0.928·8-s + (0.481 + 0.833i)9-s + (0.426 − 0.739i)10-s + (0.657 − 1.13i)11-s − 0.0240·12-s + 0.859·14-s + (−0.0774 + 0.134i)15-s + (0.554 − 0.960i)16-s + (−0.321 − 0.555i)17-s − 1.02·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.872 + 0.488i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.872 + 0.488i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(169\)    =    \(13^{2}\)
Sign: $0.872 + 0.488i$
Analytic conductor: \(9.97132\)
Root analytic conductor: \(3.15774\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{169} (22, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 169,\ (\ :3/2),\ 0.872 + 0.488i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.790100 - 0.206286i\)
\(L(\frac12)\) \(\approx\) \(0.790100 - 0.206286i\)
\(L(\frac{5}{2})\) 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 \( 1 + (1.5 - 2.59i)T + (-4 - 6.92i)T^{2} \)
3 \( 1 + (-0.5 + 0.866i)T + (-13.5 - 23.3i)T^{2} \)
5 \( 1 + 9T + 125T^{2} \)
7 \( 1 + (7.5 + 12.9i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (-24 + 41.5i)T + (-665.5 - 1.15e3i)T^{2} \)
17 \( 1 + (22.5 + 38.9i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (3 + 5.19i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-81 + 140. i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-72 + 124. i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 - 264T + 2.97e4T^{2} \)
37 \( 1 + (151.5 - 262. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (-96 + 166. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (48.5 + 84.0i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 - 111T + 1.03e5T^{2} \)
53 \( 1 + 414T + 1.48e5T^{2} \)
59 \( 1 + (261 + 452. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (188 + 325. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-18 + 31.1i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + (178.5 + 309. i)T + (-1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 + 1.09e3T + 3.89e5T^{2} \)
79 \( 1 + 830T + 4.93e5T^{2} \)
83 \( 1 + 438T + 5.71e5T^{2} \)
89 \( 1 + (-219 + 379. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (-426 - 737. i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−12.16693277848303983248110182776, −11.26139129667520852892350835824, −10.10514452258259534157879273019, −8.744426198168635485537241731664, −8.038687337217344090768644001416, −7.09115525756639724753531109780, −6.30155493213458042675539235695, −4.50659932249002245930000288101, −3.08656526809905597924963636554, −0.47705480509723496677370257319, 1.41027907882468251670775195145, 3.05774053755921450511831765440, 4.26155161376098057703570560218, 6.08866618980640591963573144870, 7.22823142167726694641313908047, 8.787242315631982601155538409272, 9.456198074146389438820582188386, 10.30346187888908419046565351261, 11.54826496835479020711723946856, 12.13248071771579872580522132964

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