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Properties

Label 2-13e2-13.2-c2-0-7
Degree $2$
Conductor $169$
Sign $0.623 + 0.781i$
Analytic cond. $4.60491$
Root an. cond. $2.14590$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−3.11 − 0.834i)2-s + (1.02 + 1.77i)3-s + (5.54 + 3.19i)4-s + (5.50 − 5.50i)5-s + (−1.70 − 6.37i)6-s + (−3.16 + 0.846i)7-s + (−5.47 − 5.47i)8-s + (2.40 − 4.16i)9-s + (−21.7 + 12.5i)10-s + (−1.31 + 4.90i)11-s + 13.0i·12-s + 10.5·14-s + (15.3 + 4.12i)15-s + (−0.322 − 0.558i)16-s + (8.72 + 5.03i)17-s + (−10.9 + 10.9i)18-s + ⋯
L(s)  = 1  + (−1.55 − 0.417i)2-s + (0.341 + 0.590i)3-s + (1.38 + 0.799i)4-s + (1.10 − 1.10i)5-s + (−0.284 − 1.06i)6-s + (−0.451 + 0.120i)7-s + (−0.683 − 0.683i)8-s + (0.267 − 0.463i)9-s + (−2.17 + 1.25i)10-s + (−0.119 + 0.445i)11-s + 1.09i·12-s + 0.753·14-s + (1.02 + 0.274i)15-s + (−0.0201 − 0.0349i)16-s + (0.513 + 0.296i)17-s + (−0.609 + 0.609i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.623 + 0.781i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.623 + 0.781i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(169\)    =    \(13^{2}\)
Sign: $0.623 + 0.781i$
Analytic conductor: \(4.60491\)
Root analytic conductor: \(2.14590\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{169} (80, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 169,\ (\ :1),\ 0.623 + 0.781i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.823796 - 0.396763i\)
\(L(\frac12)\) \(\approx\) \(0.823796 - 0.396763i\)
\(L(2)\) 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 \( 1 + (3.11 + 0.834i)T + (3.46 + 2i)T^{2} \)
3 \( 1 + (-1.02 - 1.77i)T + (-4.5 + 7.79i)T^{2} \)
5 \( 1 + (-5.50 + 5.50i)T - 25iT^{2} \)
7 \( 1 + (3.16 - 0.846i)T + (42.4 - 24.5i)T^{2} \)
11 \( 1 + (1.31 - 4.90i)T + (-104. - 60.5i)T^{2} \)
17 \( 1 + (-8.72 - 5.03i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (3.07 + 11.4i)T + (-312. + 180.5i)T^{2} \)
23 \( 1 + (-27.5 + 15.9i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (16.3 + 28.2i)T + (-420.5 + 728. i)T^{2} \)
31 \( 1 + (-8.05 + 8.05i)T - 961iT^{2} \)
37 \( 1 + (-11.4 + 42.7i)T + (-1.18e3 - 684.5i)T^{2} \)
41 \( 1 + (-45.6 - 12.2i)T + (1.45e3 + 840.5i)T^{2} \)
43 \( 1 + (-20.1 - 11.6i)T + (924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-64.6 - 64.6i)T + 2.20e3iT^{2} \)
53 \( 1 + 48.7T + 2.80e3T^{2} \)
59 \( 1 + (23.5 - 6.31i)T + (3.01e3 - 1.74e3i)T^{2} \)
61 \( 1 + (6.30 - 10.9i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-16.7 - 4.48i)T + (3.88e3 + 2.24e3i)T^{2} \)
71 \( 1 + (10.4 + 38.8i)T + (-4.36e3 + 2.52e3i)T^{2} \)
73 \( 1 + (76.5 + 76.5i)T + 5.32e3iT^{2} \)
79 \( 1 + 94.0T + 6.24e3T^{2} \)
83 \( 1 + (33.6 - 33.6i)T - 6.88e3iT^{2} \)
89 \( 1 + (9.39 - 35.0i)T + (-6.85e3 - 3.96e3i)T^{2} \)
97 \( 1 + (-10.7 - 40.1i)T + (-8.14e3 + 4.70e3i)T^{2} \)
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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

−12.39893746351157645278052581707, −10.92997551735630090469630904665, −9.970114589987452088552911722058, −9.263098967936142922475680428269, −8.998022855513645814887861141494, −7.62457284787183206338454480440, −6.14296158577976590342652459362, −4.57935272577756933731728372391, −2.57656874567734805831315343369, −1.01445487564334024486990373070, 1.51557483897666923047778109158, 2.89347416876358866538589217714, 5.75372370587984802745357989225, 6.86699348123274564144303927029, 7.41181466711533650816275581904, 8.613028364233845104979939442840, 9.675886173425645541891415449394, 10.35410080224383763300898454572, 11.12915140852926141118624692469, 12.88715852891347977150662047081

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