Properties

Label 2-13e2-169.25-c1-0-0
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: $\chi_{169} (25, \cdot )$
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

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

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