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Properties

Label 2-13e2-13.3-c3-0-25
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  + (−1.28 − 2.21i)2-s + (1.84 + 3.19i)3-s + (0.719 − 1.24i)4-s + 0.561·5-s + (4.71 − 8.17i)6-s + (−9.08 + 15.7i)7-s − 24.1·8-s + (6.71 − 11.6i)9-s + (−0.719 − 1.24i)10-s + (−32.3 − 56.0i)11-s + 5.30·12-s + 46.5·14-s + (1.03 + 1.79i)15-s + (25.2 + 43.6i)16-s + (12.7 − 22.1i)17-s − 34.3·18-s + ⋯
L(s)  = 1  + (−0.452 − 0.784i)2-s + (0.354 + 0.614i)3-s + (0.0899 − 0.155i)4-s + 0.0502·5-s + (0.321 − 0.556i)6-s + (−0.490 + 0.849i)7-s − 1.06·8-s + (0.248 − 0.430i)9-s + (−0.0227 − 0.0393i)10-s + (−0.887 − 1.53i)11-s + 0.127·12-s + 0.888·14-s + (0.0178 + 0.0308i)15-s + (0.393 + 0.682i)16-s + (0.182 − 0.315i)17-s − 0.450·18-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: $\chi_{169} (146, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 169,\ (\ :3/2),\ -0.872 + 0.488i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.231529 - 0.886783i\)
\(L(\frac12)\) \(\approx\) \(0.231529 - 0.886783i\)
\(L(\frac{5}{2})\) 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 \( 1 + (1.28 + 2.21i)T + (-4 + 6.92i)T^{2} \)
3 \( 1 + (-1.84 - 3.19i)T + (-13.5 + 23.3i)T^{2} \)
5 \( 1 - 0.561T + 125T^{2} \)
7 \( 1 + (9.08 - 15.7i)T + (-171.5 - 297. i)T^{2} \)
11 \( 1 + (32.3 + 56.0i)T + (-665.5 + 1.15e3i)T^{2} \)
17 \( 1 + (-12.7 + 22.1i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-53.9 + 93.5i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (36.6 + 63.4i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (87.9 + 152. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + 113.T + 2.97e4T^{2} \)
37 \( 1 + (57.4 + 99.4i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (-34.8 - 60.3i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (219. - 379. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + 31.9T + 1.03e5T^{2} \)
53 \( 1 - 2.84T + 1.48e5T^{2} \)
59 \( 1 + (35.8 - 62.0i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-460. + 797. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-222. - 384. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + (-270. + 469. i)T + (-1.78e5 - 3.09e5i)T^{2} \)
73 \( 1 - 764.T + 3.89e5T^{2} \)
79 \( 1 + 421.T + 4.93e5T^{2} \)
83 \( 1 - 603.T + 5.71e5T^{2} \)
89 \( 1 + (-579. - 1.00e3i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (291. - 505. i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−11.61497385325112472318886278447, −10.88584368082938731139863261068, −9.724504376464827791275988995464, −9.270299142840876560129056485539, −8.208238680157439429258784355755, −6.37051195911218099393106461508, −5.39699266518487194503548511801, −3.44947885884092282924320216215, −2.54321628636650256424530971474, −0.44194807365249133003897886939, 1.95652601494146955328175586963, 3.65808196524719961611417011260, 5.44835332201661153865513189541, 6.99527308058620390429088172851, 7.42982581603841182947961645868, 8.195272314577383850875855156911, 9.671714918396222049874846998149, 10.40122851724185191646898306236, 12.05048192882306585109859818240, 12.80610647486775650638328882253

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