Properties

Label 2-13e2-13.4-c1-0-3
Degree $2$
Conductor $169$
Sign $0.756 - 0.654i$
Analytic cond. $1.34947$
Root an. cond. $1.16166$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.94 + 1.12i)2-s + (0.277 − 0.480i)3-s + (1.52 + 2.64i)4-s − 1.44i·5-s + (1.07 − 0.623i)6-s + (−1.77 + 1.02i)7-s + 2.35i·8-s + (1.34 + 2.33i)9-s + (1.62 − 2.81i)10-s + (−2.21 − 1.27i)11-s + 1.69·12-s − 4.60·14-s + (−0.694 − 0.400i)15-s + (0.400 − 0.694i)16-s + (−2.64 − 4.58i)17-s + 6.04i·18-s + ⋯
L(s)  = 1  + (1.37 + 0.794i)2-s + (0.160 − 0.277i)3-s + (0.762 + 1.32i)4-s − 0.646i·5-s + (0.440 − 0.254i)6-s + (−0.670 + 0.387i)7-s + 0.833i·8-s + (0.448 + 0.777i)9-s + (0.513 − 0.889i)10-s + (−0.667 − 0.385i)11-s + 0.488·12-s − 1.23·14-s + (−0.179 − 0.103i)15-s + (0.100 − 0.173i)16-s + (−0.642 − 1.11i)17-s + 1.42i·18-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(169\)    =    \(13^{2}\)
Sign: $0.756 - 0.654i$
Analytic conductor: \(1.34947\)
Root analytic conductor: \(1.16166\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{169} (147, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 169,\ (\ :1/2),\ 0.756 - 0.654i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.03891 + 0.759446i\)
\(L(\frac12)\) \(\approx\) \(2.03891 + 0.759446i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.94 - 1.12i)T + (1 + 1.73i)T^{2} \)
3 \( 1 + (-0.277 + 0.480i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + 1.44iT - 5T^{2} \)
7 \( 1 + (1.77 - 1.02i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (2.21 + 1.27i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (2.64 + 4.58i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (5.06 - 2.92i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.945 - 1.63i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.13 - 1.96i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 4.26iT - 31T^{2} \)
37 \( 1 + (-4.63 - 2.67i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.10 + 0.637i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.06 - 5.31i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 2.95iT - 47T^{2} \)
53 \( 1 - 5.52T + 53T^{2} \)
59 \( 1 + (-10.5 + 6.10i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.28 + 7.41i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.499 + 0.288i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (3.97 - 2.29i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 10.5iT - 73T^{2} \)
79 \( 1 + 15.7T + 79T^{2} \)
83 \( 1 - 7.72iT - 83T^{2} \)
89 \( 1 + (-5.72 - 3.30i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-10.3 + 5.96i)T + (48.5 - 84.0i)T^{2} \)
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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

−12.96298654111264542757518938454, −12.61502603130588421403566559186, −11.26274671183776907529780466152, −9.819896595905966002434392372022, −8.461313154885485783024857528929, −7.42903311837610376939957939900, −6.33104242306520902182434141640, −5.27892156123455149252406889098, −4.31880840962357660936694171757, −2.69913533684639126718970353393, 2.43773171356557156339063238417, 3.65858707052145433046522796296, 4.52059002084540638853318211425, 6.11094955353600163022289963231, 6.97331508691297240533172706120, 8.790825493988043063306844503415, 10.35741081118551182658961957974, 10.56999184520355058507794007850, 11.90280646700288506450710025500, 12.93917589944882892511965521423

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