Properties

Label 2-13e2-13.3-c3-0-15
Degree $2$
Conductor $169$
Sign $0.664 + 0.746i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.664 + 0.746i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.664 + 0.746i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(169\)    =    \(13^{2}\)
Sign: $0.664 + 0.746i$
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: \((2,\ 169,\ (\ :3/2),\ 0.664 + 0.746i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.84727 - 0.828701i\)
\(L(\frac12)\) \(\approx\) \(1.84727 - 0.828701i\)
\(L(\frac{5}{2})\) 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

−11.81228725481819621235632891538, −10.70844899207600945582929921921, −10.16224787286871200564989302527, −9.450984162130675076569323746027, −8.783186784655053790692296812578, −7.10221872891256214390298377305, −5.35242030642084998889721545685, −4.03323144945382492288956548147, −2.69994388055419531329030054709, −1.31248309036038851134004631207, 1.53757732922171221992105764468, 2.75587944190217775525305487949, 5.57835254209185426275708712413, 6.21017703840753424181191599403, 7.34105417824589730900377208084, 8.367453648465941067597285637160, 8.852435582945891632986195043265, 9.982581888653905024977317922823, 11.73669423240321516893975729681, 12.60349263982830510044770485056

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