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Properties

Label 2-13e2-13.4-c1-0-0
Degree $2$
Conductor $169$
Sign $-0.000427 - 0.999i$
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  + (−0.480 − 0.277i)2-s + (−0.400 + 0.694i)3-s + (−0.846 − 1.46i)4-s + 2.80i·5-s + (0.385 − 0.222i)6-s + (−2.33 + 1.34i)7-s + 2.04i·8-s + (1.17 + 2.04i)9-s + (0.777 − 1.34i)10-s + (1.03 + 0.599i)11-s + 1.35·12-s + 1.49·14-s + (−1.94 − 1.12i)15-s + (−1.12 + 1.94i)16-s + (0.568 + 0.984i)17-s − 1.30i·18-s + ⋯
L(s)  = 1  + (−0.339 − 0.196i)2-s + (−0.231 + 0.400i)3-s + (−0.423 − 0.732i)4-s + 1.25i·5-s + (0.157 − 0.0908i)6-s + (−0.881 + 0.508i)7-s + 0.724i·8-s + (0.392 + 0.680i)9-s + (0.245 − 0.425i)10-s + (0.312 + 0.180i)11-s + 0.391·12-s + 0.399·14-s + (−0.502 − 0.290i)15-s + (−0.280 + 0.486i)16-s + (0.137 + 0.238i)17-s − 0.308i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.000427 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.000427 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(169\)    =    \(13^{2}\)
Sign: $-0.000427 - 0.999i$
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: \((2,\ 169,\ (\ :1/2),\ -0.000427 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.469041 + 0.469242i\)
\(L(\frac12)\) \(\approx\) \(0.469041 + 0.469242i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.480 + 0.277i)T + (1 + 1.73i)T^{2} \)
3 \( 1 + (0.400 - 0.694i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 - 2.80iT - 5T^{2} \)
7 \( 1 + (2.33 - 1.34i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.03 - 0.599i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (-0.568 - 0.984i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.67 - 0.969i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.30 - 3.98i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.94 + 6.84i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 5.89iT - 31T^{2} \)
37 \( 1 + (0.823 + 0.475i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-2.87 - 1.65i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.57 - 6.19i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 7.69iT - 47T^{2} \)
53 \( 1 - 5.87T + 53T^{2} \)
59 \( 1 + (0.0104 - 0.00604i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.01 - 6.96i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (8.01 + 4.62i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-11.9 + 6.87i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 12.8iT - 73T^{2} \)
79 \( 1 - 0.807T + 79T^{2} \)
83 \( 1 - 16.3iT - 83T^{2} \)
89 \( 1 + (-12.7 - 7.36i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (2.71 - 1.56i)T + (48.5 - 84.0i)T^{2} \)
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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

−13.16167123803830452155939542773, −11.72342916861155124506687739338, −10.78365389824858166279144138871, −10.03275311895906625248981338163, −9.447430140202594236040172649331, −7.938388278095048654047493494756, −6.52852347987941915441397577208, −5.64925991492377957926501827586, −4.08603560077506627328495444489, −2.36923133843835344338977762154, 0.74000514482877958804681191397, 3.55561964952886885861486232108, 4.70057707117845540193698907987, 6.40460149604969715917897252329, 7.30336784755370846907168521622, 8.632128244739303062441347219644, 9.188493345319291688556141589774, 10.29611361959847873725704718325, 12.03915511191344637703860974337, 12.58553962031855460535122894442

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