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Properties

Label 2-13e2-13.6-c2-0-12
Degree $2$
Conductor $169$
Sign $0.490 - 0.871i$
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  + (0.834 + 3.11i)2-s + (1.02 − 1.77i)3-s + (−5.54 + 3.19i)4-s + (5.50 − 5.50i)5-s + (6.37 + 1.70i)6-s + (0.846 − 3.16i)7-s + (−5.47 − 5.47i)8-s + (2.40 + 4.16i)9-s + (21.7 + 12.5i)10-s + (4.90 − 1.31i)11-s + 13.0i·12-s + 10.5·14-s + (−4.12 − 15.3i)15-s + (−0.322 + 0.558i)16-s + (−8.72 + 5.03i)17-s + (−10.9 + 10.9i)18-s + ⋯
L(s)  = 1  + (0.417 + 1.55i)2-s + (0.341 − 0.590i)3-s + (−1.38 + 0.799i)4-s + (1.10 − 1.10i)5-s + (1.06 + 0.284i)6-s + (0.120 − 0.451i)7-s + (−0.683 − 0.683i)8-s + (0.267 + 0.463i)9-s + (2.17 + 1.25i)10-s + (0.445 − 0.119i)11-s + 1.09i·12-s + 0.753·14-s + (−0.274 − 1.02i)15-s + (−0.0201 + 0.0349i)16-s + (−0.513 + 0.296i)17-s + (−0.609 + 0.609i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.490 - 0.871i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.490 - 0.871i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(169\)    =    \(13^{2}\)
Sign: $0.490 - 0.871i$
Analytic conductor: \(4.60491\)
Root analytic conductor: \(2.14590\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{169} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 169,\ (\ :1),\ 0.490 - 0.871i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.99157 + 1.16391i\)
\(L(\frac12)\) \(\approx\) \(1.99157 + 1.16391i\)
\(L(2)\) 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 \( 1 + (-0.834 - 3.11i)T + (-3.46 + 2i)T^{2} \)
3 \( 1 + (-1.02 + 1.77i)T + (-4.5 - 7.79i)T^{2} \)
5 \( 1 + (-5.50 + 5.50i)T - 25iT^{2} \)
7 \( 1 + (-0.846 + 3.16i)T + (-42.4 - 24.5i)T^{2} \)
11 \( 1 + (-4.90 + 1.31i)T + (104. - 60.5i)T^{2} \)
17 \( 1 + (8.72 - 5.03i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (-11.4 - 3.07i)T + (312. + 180.5i)T^{2} \)
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 \( 1 + (42.7 - 11.4i)T + (1.18e3 - 684.5i)T^{2} \)
41 \( 1 + (12.2 + 45.6i)T + (-1.45e3 + 840.5i)T^{2} \)
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 \( 1 + (-6.31 + 23.5i)T + (-3.01e3 - 1.74e3i)T^{2} \)
61 \( 1 + (6.30 + 10.9i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (4.48 + 16.7i)T + (-3.88e3 + 2.24e3i)T^{2} \)
71 \( 1 + (-38.8 - 10.4i)T + (4.36e3 + 2.52e3i)T^{2} \)
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 \( 1 + (-35.0 + 9.39i)T + (6.85e3 - 3.96e3i)T^{2} \)
97 \( 1 + (40.1 + 10.7i)T + (8.14e3 + 4.70e3i)T^{2} \)
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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

−13.12363085828830677844577161897, −12.37738412800966690136172860212, −10.49743896082641954396176264768, −9.163813845877799442627957628893, −8.370887297134837583994183269813, −7.39818970242977387096362096528, −6.35040370591242675060608885704, −5.35635499153549561022006844906, −4.33960764371974334111597651484, −1.69444206661873671969610940591, 1.89328577543986820497774012954, 2.99946770402706118679049980683, 4.07626633723213934618286624932, 5.59364696846359361172676593347, 6.92949089573850291387564904884, 8.992530340485874255169856447400, 9.832299557686422145198962827002, 10.25541217210852889923452372454, 11.41245086852986344249940947949, 12.13038209555040178120604894102

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