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Properties

Label 2-13e2-13.12-c1-0-6
Degree 22
Conductor 169169
Sign 0.691+0.722i0.691 + 0.722i
Analytic cond. 1.349471.34947
Root an. cond. 1.161661.16166
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.554i·2-s + 0.801·3-s + 1.69·4-s − 2.80i·5-s − 0.445i·6-s + 2.69i·7-s − 2.04i·8-s − 2.35·9-s − 1.55·10-s + 1.19i·11-s + 1.35·12-s + 1.49·14-s − 2.24i·15-s + 2.24·16-s − 1.13·17-s + 1.30i·18-s + ⋯
L(s)  = 1  − 0.392i·2-s + 0.462·3-s + 0.846·4-s − 1.25i·5-s − 0.181i·6-s + 1.01i·7-s − 0.724i·8-s − 0.785·9-s − 0.491·10-s + 0.361i·11-s + 0.391·12-s + 0.399·14-s − 0.580i·15-s + 0.561·16-s − 0.275·17-s + 0.308i·18-s + ⋯

Functional equation

Λ(s)=(169s/2 ΓC(s) L(s)=((0.691+0.722i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.691 + 0.722i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(169s/2 ΓC(s+1/2) L(s)=((0.691+0.722i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.691 + 0.722i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 169169    =    13213^{2}
Sign: 0.691+0.722i0.691 + 0.722i
Analytic conductor: 1.349471.34947
Root analytic conductor: 1.161661.16166
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ169(168,⋅)\chi_{169} (168, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 169, ( :1/2), 0.691+0.722i)(2,\ 169,\ (\ :1/2),\ 0.691 + 0.722i)

Particular Values

L(1)L(1) ≈\approx 1.35538−0.578613i1.35538 - 0.578613i
L(12)L(\frac12) ≈\approx 1.35538−0.578613i1.35538 - 0.578613i
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad13 1 1
good2 1+0.554iT−2T2 1 + 0.554iT - 2T^{2}
3 1−0.801T+3T2 1 - 0.801T + 3T^{2}
5 1+2.80iT−5T2 1 + 2.80iT - 5T^{2}
7 1−2.69iT−7T2 1 - 2.69iT - 7T^{2}
11 1−1.19iT−11T2 1 - 1.19iT - 11T^{2}
17 1+1.13T+17T2 1 + 1.13T + 17T^{2}
19 1−1.93iT−19T2 1 - 1.93iT - 19T^{2}
23 1−4.60T+23T2 1 - 4.60T + 23T^{2}
29 1+7.89T+29T2 1 + 7.89T + 29T^{2}
31 1−5.89iT−31T2 1 - 5.89iT - 31T^{2}
37 1+0.951iT−37T2 1 + 0.951iT - 37T^{2}
41 1−3.31iT−41T2 1 - 3.31iT - 41T^{2}
43 1+7.15T+43T2 1 + 7.15T + 43T^{2}
47 1−7.69iT−47T2 1 - 7.69iT - 47T^{2}
53 1−5.87T+53T2 1 - 5.87T + 53T^{2}
59 1−0.0120iT−59T2 1 - 0.0120iT - 59T^{2}
61 1+8.03T+61T2 1 + 8.03T + 61T^{2}
67 1+9.25iT−67T2 1 + 9.25iT - 67T^{2}
71 1+13.7iT−71T2 1 + 13.7iT - 71T^{2}
73 1+12.8iT−73T2 1 + 12.8iT - 73T^{2}
79 1−0.807T+79T2 1 - 0.807T + 79T^{2}
83 1+16.3iT−83T2 1 + 16.3iT - 83T^{2}
89 1−14.7iT−89T2 1 - 14.7iT - 89T^{2}
97 1−3.13iT−97T2 1 - 3.13iT - 97T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(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

−12.46411806192441046530082690304, −11.86302020321111432584732948672, −10.83927107409341649636334428678, −9.375482347468606611056271135866, −8.778823233459227880407182999975, −7.66471625855674818277110366204, −6.12411235041285051012484329717, −5.02991880792359541495104972319, −3.21612061595007245431824799924, −1.85363386397674879286724229876, 2.50507082103092567003728858686, 3.58631967682433352224547082499, 5.67399485917868166993388689587, 6.86912401101177441015696700849, 7.41409192544279426129151621974, 8.614244742138536684828859967988, 10.10481225846461631473521577550, 11.10911071766157646178444943363, 11.42699065598586314950194827702, 13.23172843026177316489265010637

Graph of the ZZ-function along the critical line