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Properties

Label 2-13e2-13.4-c1-0-4
Degree $2$
Conductor $169$
Sign $0.743 + 0.669i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.743 + 0.669i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.743 + 0.669i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(169\)    =    \(13^{2}\)
Sign: $0.743 + 0.669i$
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: \((2,\ 169,\ (\ :1/2),\ 0.743 + 0.669i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.14682 - 0.440147i\)
\(L(\frac12)\) \(\approx\) \(1.14682 - 0.440147i\)
\(L(\frac{3}{2})\) 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

−12.94967716557887995642779659682, −11.65758182800165973462589457315, −10.55878793468737309613265154226, −9.761686223817259963896891061791, −8.607890823286435352222785755496, −7.57096896499265022495283134828, −5.78801193016380298781070881330, −4.88530157431531357726722964538, −4.29754343888588233290779195926, −1.31166572087451484341210837336, 2.44663173544988834750011041602, 3.78534968064688741775234395804, 5.24005039950589180306815428430, 6.68273889576667765495275321307, 7.61814349759604108206015689597, 8.669893062389116962473054578818, 10.04852793985339106315690177065, 11.21991671454555849680413373551, 11.97073548208706718943451216001, 12.71616402726431068553355294366

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