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Properties

Label 2-13e2-13.9-c1-0-1
Degree 22
Conductor 169169
Sign −0.434−0.900i-0.434 - 0.900i
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  + (−1.12 + 1.94i)2-s + (0.277 − 0.480i)3-s + (−1.52 − 2.64i)4-s + 1.44·5-s + (0.623 + 1.07i)6-s + (1.02 + 1.77i)7-s + 2.35·8-s + (1.34 + 2.33i)9-s + (−1.62 + 2.81i)10-s + (−1.27 + 2.21i)11-s − 1.69·12-s − 4.60·14-s + (0.400 − 0.694i)15-s + (0.400 − 0.694i)16-s + (2.64 + 4.58i)17-s − 6.04·18-s + ⋯
L(s)  = 1  + (−0.794 + 1.37i)2-s + (0.160 − 0.277i)3-s + (−0.762 − 1.32i)4-s + 0.646·5-s + (0.254 + 0.440i)6-s + (0.387 + 0.670i)7-s + 0.833·8-s + (0.448 + 0.777i)9-s + (−0.513 + 0.889i)10-s + (−0.385 + 0.667i)11-s − 0.488·12-s − 1.23·14-s + (0.103 − 0.179i)15-s + (0.100 − 0.173i)16-s + (0.642 + 1.11i)17-s − 1.42·18-s + ⋯

Functional equation

Λ(s)=(169s/2 ΓC(s) L(s)=((−0.434−0.900i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.434 - 0.900i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(169s/2 ΓC(s+1/2) L(s)=((−0.434−0.900i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.434 - 0.900i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 169169    =    13213^{2}
Sign: −0.434−0.900i-0.434 - 0.900i
Analytic conductor: 1.349471.34947
Root analytic conductor: 1.161661.16166
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ169(22,⋅)\chi_{169} (22, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 169, ( :1/2), −0.434−0.900i)(2,\ 169,\ (\ :1/2),\ -0.434 - 0.900i)

Particular Values

L(1)L(1) ≈\approx 0.468862+0.746537i0.468862 + 0.746537i
L(12)L(\frac12) ≈\approx 0.468862+0.746537i0.468862 + 0.746537i
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+(1.12−1.94i)T+(−1−1.73i)T2 1 + (1.12 - 1.94i)T + (-1 - 1.73i)T^{2}
3 1+(−0.277+0.480i)T+(−1.5−2.59i)T2 1 + (-0.277 + 0.480i)T + (-1.5 - 2.59i)T^{2}
5 1−1.44T+5T2 1 - 1.44T + 5T^{2}
7 1+(−1.02−1.77i)T+(−3.5+6.06i)T2 1 + (-1.02 - 1.77i)T + (-3.5 + 6.06i)T^{2}
11 1+(1.27−2.21i)T+(−5.5−9.52i)T2 1 + (1.27 - 2.21i)T + (-5.5 - 9.52i)T^{2}
17 1+(−2.64−4.58i)T+(−8.5+14.7i)T2 1 + (-2.64 - 4.58i)T + (-8.5 + 14.7i)T^{2}
19 1+(2.92+5.06i)T+(−9.5+16.4i)T2 1 + (2.92 + 5.06i)T + (-9.5 + 16.4i)T^{2}
23 1+(−0.945+1.63i)T+(−11.5−19.9i)T2 1 + (-0.945 + 1.63i)T + (-11.5 - 19.9i)T^{2}
29 1+(1.13−1.96i)T+(−14.5−25.1i)T2 1 + (1.13 - 1.96i)T + (-14.5 - 25.1i)T^{2}
31 1−4.26T+31T2 1 - 4.26T + 31T^{2}
37 1+(−2.67+4.63i)T+(−18.5−32.0i)T2 1 + (-2.67 + 4.63i)T + (-18.5 - 32.0i)T^{2}
41 1+(−0.637+1.10i)T+(−20.5−35.5i)T2 1 + (-0.637 + 1.10i)T + (-20.5 - 35.5i)T^{2}
43 1+(3.06+5.31i)T+(−21.5+37.2i)T2 1 + (3.06 + 5.31i)T + (-21.5 + 37.2i)T^{2}
47 1−2.95T+47T2 1 - 2.95T + 47T^{2}
53 1−5.52T+53T2 1 - 5.52T + 53T^{2}
59 1+(6.10+10.5i)T+(−29.5+51.0i)T2 1 + (6.10 + 10.5i)T + (-29.5 + 51.0i)T^{2}
61 1+(4.28+7.41i)T+(−30.5+52.8i)T2 1 + (4.28 + 7.41i)T + (-30.5 + 52.8i)T^{2}
67 1+(−0.288+0.499i)T+(−33.5−58.0i)T2 1 + (-0.288 + 0.499i)T + (-33.5 - 58.0i)T^{2}
71 1+(2.29+3.97i)T+(−35.5+61.4i)T2 1 + (2.29 + 3.97i)T + (-35.5 + 61.4i)T^{2}
73 1−10.5T+73T2 1 - 10.5T + 73T^{2}
79 1+15.7T+79T2 1 + 15.7T + 79T^{2}
83 1+7.72T+83T2 1 + 7.72T + 83T^{2}
89 1+(−3.30+5.72i)T+(−44.5−77.0i)T2 1 + (-3.30 + 5.72i)T + (-44.5 - 77.0i)T^{2}
97 1+(−5.96−10.3i)T+(−48.5+84.0i)T2 1 + (-5.96 - 10.3i)T + (-48.5 + 84.0i)T^{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

−13.30274802493991943642062769430, −12.34513401850382240147921692645, −10.70270267797774437592931441116, −9.789333694698740747997270509197, −8.734142681309732121573547923881, −7.945879412669335960905066978024, −6.98069586484077288246712179424, −5.88007385283961264644554586828, −4.87754023786801768500699109148, −2.10733452583127623776427609752, 1.21457931513965198247650443666, 2.92403365456373249923848739359, 4.17350699102445290711291341779, 5.98004456164209953798977206940, 7.66314262783182347160004044771, 8.773825595905640774039793309036, 9.854957865324813564497159491176, 10.19863433196619759390761616131, 11.33204111691160329989223093198, 12.15836106979494755575747082313

Graph of the ZZ-function along the critical line