Properties

Label 2-13e2-169.142-c1-0-9
Degree $2$
Conductor $169$
Sign $-0.288 + 0.957i$
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  + (−1.49 + 0.567i)2-s + (1.18 − 1.71i)3-s + (0.417 − 0.369i)4-s + (−2.96 + 0.359i)5-s + (−0.798 + 3.24i)6-s + (1.24 − 2.37i)7-s + (1.07 − 2.04i)8-s + (−0.481 − 1.26i)9-s + (4.22 − 2.21i)10-s + (−4.01 − 1.52i)11-s + (−0.140 − 1.15i)12-s + (−1.01 − 3.45i)13-s + (−0.516 + 4.25i)14-s + (−2.89 + 5.51i)15-s + (−0.579 + 4.76i)16-s + (−3.38 − 1.77i)17-s + ⋯
L(s)  = 1  + (−1.05 + 0.400i)2-s + (0.684 − 0.991i)3-s + (0.208 − 0.184i)4-s + (−1.32 + 0.160i)5-s + (−0.326 + 1.32i)6-s + (0.470 − 0.897i)7-s + (0.379 − 0.722i)8-s + (−0.160 − 0.423i)9-s + (1.33 − 0.701i)10-s + (−1.20 − 0.458i)11-s + (−0.0404 − 0.333i)12-s + (−0.281 − 0.959i)13-s + (−0.138 + 1.13i)14-s + (−0.747 + 1.42i)15-s + (−0.144 + 1.19i)16-s + (−0.820 − 0.430i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.288 + 0.957i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.288 + 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(169\)    =    \(13^{2}\)
Sign: $-0.288 + 0.957i$
Analytic conductor: \(1.34947\)
Root analytic conductor: \(1.16166\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{169} (142, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 169,\ (\ :1/2),\ -0.288 + 0.957i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.300798 - 0.404813i\)
\(L(\frac12)\) \(\approx\) \(0.300798 - 0.404813i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 + (1.01 + 3.45i)T \)
good2 \( 1 + (1.49 - 0.567i)T + (1.49 - 1.32i)T^{2} \)
3 \( 1 + (-1.18 + 1.71i)T + (-1.06 - 2.80i)T^{2} \)
5 \( 1 + (2.96 - 0.359i)T + (4.85 - 1.19i)T^{2} \)
7 \( 1 + (-1.24 + 2.37i)T + (-3.97 - 5.76i)T^{2} \)
11 \( 1 + (4.01 + 1.52i)T + (8.23 + 7.29i)T^{2} \)
17 \( 1 + (3.38 + 1.77i)T + (9.65 + 13.9i)T^{2} \)
19 \( 1 + 3.49iT - 19T^{2} \)
23 \( 1 - 6.33T + 23T^{2} \)
29 \( 1 + (1.14 + 3.01i)T + (-21.7 + 19.2i)T^{2} \)
31 \( 1 + (1.63 - 6.62i)T + (-27.4 - 14.4i)T^{2} \)
37 \( 1 + (-0.694 + 2.81i)T + (-32.7 - 17.1i)T^{2} \)
41 \( 1 + (-6.02 - 4.15i)T + (14.5 + 38.3i)T^{2} \)
43 \( 1 + (-7.11 + 1.75i)T + (38.0 - 19.9i)T^{2} \)
47 \( 1 + (6.61 - 7.46i)T + (-5.66 - 46.6i)T^{2} \)
53 \( 1 + (5.07 + 2.66i)T + (30.1 + 43.6i)T^{2} \)
59 \( 1 + (-13.8 + 1.68i)T + (57.2 - 14.1i)T^{2} \)
61 \( 1 + (1.93 - 1.01i)T + (34.6 - 50.2i)T^{2} \)
67 \( 1 + (-3.89 + 4.40i)T + (-8.07 - 66.5i)T^{2} \)
71 \( 1 + (6.65 + 4.59i)T + (25.1 + 66.3i)T^{2} \)
73 \( 1 + (-1.17 - 0.445i)T + (54.6 + 48.4i)T^{2} \)
79 \( 1 + (8.90 + 7.88i)T + (9.52 + 78.4i)T^{2} \)
83 \( 1 + (1.30 - 0.903i)T + (29.4 - 77.6i)T^{2} \)
89 \( 1 + 12.0iT - 89T^{2} \)
97 \( 1 + (-9.94 - 1.20i)T + (94.1 + 23.2i)T^{2} \)
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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

−12.88098426020563900940310309485, −11.18633421600418642334231892238, −10.56283125313973816115023426139, −8.962863379364863356513637276833, −8.025018218087133103838914208372, −7.59814521345332135552112016932, −7.00156164955688496591522028349, −4.64658841004868605683171109816, −3.01216817725829301916497729411, −0.61931833794579224025595275487, 2.41218962166345106848897772604, 4.11630758341286690231004757641, 5.10132694858033850282924500273, 7.45097932499313866579197808786, 8.406550609258147456601582098943, 8.954408560519182509054646762254, 9.902305974911855808982007898470, 10.94766242652921337722291449711, 11.66239981358110223097956661488, 12.83890203418770434358269250108

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