Properties

Label 2-13e2-13.4-c3-0-28
Degree $2$
Conductor $169$
Sign $-0.756 + 0.654i$
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  + (−0.129 − 0.0745i)2-s + (3.24 − 5.61i)3-s + (−3.98 − 6.90i)4-s − 10.2i·5-s + (−0.837 + 0.483i)6-s + (25.6 − 14.8i)7-s + 2.38i·8-s + (−7.55 − 13.0i)9-s + (−0.764 + 1.32i)10-s + (32.9 + 19.0i)11-s − 51.7·12-s − 4.42·14-s + (−57.6 − 33.2i)15-s + (−31.7 + 54.9i)16-s + (−35.6 − 61.7i)17-s + 2.25i·18-s + ⋯
L(s)  = 1  + (−0.0456 − 0.0263i)2-s + (0.624 − 1.08i)3-s + (−0.498 − 0.863i)4-s − 0.917i·5-s + (−0.0569 + 0.0329i)6-s + (1.38 − 0.801i)7-s + 0.105i·8-s + (−0.279 − 0.484i)9-s + (−0.0241 + 0.0418i)10-s + (0.904 + 0.522i)11-s − 1.24·12-s − 0.0844·14-s + (−0.991 − 0.572i)15-s + (−0.495 + 0.858i)16-s + (−0.508 − 0.880i)17-s + 0.0294i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.756 + 0.654i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.756 + 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(169\)    =    \(13^{2}\)
Sign: $-0.756 + 0.654i$
Analytic conductor: \(9.97132\)
Root analytic conductor: \(3.15774\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{169} (147, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 169,\ (\ :3/2),\ -0.756 + 0.654i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.732233 - 1.96585i\)
\(L(\frac12)\) \(\approx\) \(0.732233 - 1.96585i\)
\(L(\frac{5}{2})\) 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 \( 1 + (0.129 + 0.0745i)T + (4 + 6.92i)T^{2} \)
3 \( 1 + (-3.24 + 5.61i)T + (-13.5 - 23.3i)T^{2} \)
5 \( 1 + 10.2iT - 125T^{2} \)
7 \( 1 + (-25.6 + 14.8i)T + (171.5 - 297. i)T^{2} \)
11 \( 1 + (-32.9 - 19.0i)T + (665.5 + 1.15e3i)T^{2} \)
17 \( 1 + (35.6 + 61.7i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-8.74 + 5.04i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (99.3 - 172. i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (15.4 - 26.6i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 - 151. iT - 2.97e4T^{2} \)
37 \( 1 + (-131. - 75.6i)T + (2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (-179. - 103. i)T + (3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (151. + 262. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + 12.2iT - 1.03e5T^{2} \)
53 \( 1 + 250.T + 1.48e5T^{2} \)
59 \( 1 + (338. - 195. i)T + (1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-78.2 - 135. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-263. - 151. i)T + (1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + (-791. + 456. i)T + (1.78e5 - 3.09e5i)T^{2} \)
73 \( 1 - 249. iT - 3.89e5T^{2} \)
79 \( 1 + 147.T + 4.93e5T^{2} \)
83 \( 1 - 1.02e3iT - 5.71e5T^{2} \)
89 \( 1 + (819. + 473. i)T + (3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (-361. + 208. i)T + (4.56e5 - 7.90e5i)T^{2} \)
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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

−12.03378104909499523083914156836, −11.05419996773245439752938032603, −9.676116318582253397455254591493, −8.762844045657744625347616597403, −7.85235640273387391271302555237, −6.86576776287021268852537204725, −5.20128529922335189377150849297, −4.31554061166437277441795375856, −1.73169256558626728322960186368, −1.07740028819914342822537702971, 2.49401839899275658640763122227, 3.80061060701876548432138402179, 4.62447299064145182443891764700, 6.33908299709976001088073610490, 7.992534130041682201391407232707, 8.607027973611158171612734246680, 9.474516483353598058313136051879, 10.75281319380800788671317902842, 11.54217835582285221881392976597, 12.61881154291462370615063698534

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