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Properties

Label 2-13e2-13.3-c3-0-14
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  + (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

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

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

\(L(2)\) \(\approx\) \(2.49817 + 1.48535i\)
\(L(\frac12)\) \(\approx\) \(2.49817 + 1.48535i\)
\(L(\frac{5}{2})\) 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

−12.63229703504498451132815061367, −11.50508597461312188629937690914, −10.11441655767925697466376840473, −9.783738113085427112857862486333, −8.102330198186100131230183158952, −7.13373958230152592505366690610, −6.18742829622755148770092677614, −4.53755512544932741186045453364, −4.07775824836160001993222480800, −1.61847531322624449971105759598, 1.52746612954203805817548679790, 2.69148743099047099217927677755, 3.99827115024216889201159326961, 5.51985043016919928253989513626, 6.95788491909068058194357041145, 8.144534801621012939231831879453, 8.873181406561208984578249387727, 10.53823595988668744785082861257, 11.48161366972140751389020992419, 12.01521097281485440861304588993

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