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Properties

Label 2-13e2-169.10-c1-0-9
Degree 22
Conductor 169169
Sign 0.369+0.929i0.369 + 0.929i
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  + (−2.52 − 0.514i)2-s + (2.70 − 0.439i)3-s + (4.24 + 1.81i)4-s + (0.847 − 3.43i)5-s + (−7.05 − 0.284i)6-s + (0.693 − 0.328i)7-s + (−5.54 − 3.82i)8-s + (4.28 − 1.43i)9-s + (−3.90 + 8.22i)10-s + (−1.32 + 3.95i)11-s + (12.2 + 3.03i)12-s + (−2.08 + 2.94i)13-s + (−1.91 + 0.472i)14-s + (0.781 − 9.67i)15-s + (5.60 + 5.83i)16-s + (−2.29 − 4.83i)17-s + ⋯
L(s)  = 1  + (−1.78 − 0.363i)2-s + (1.56 − 0.254i)3-s + (2.12 + 0.905i)4-s + (0.378 − 1.53i)5-s + (−2.87 − 0.115i)6-s + (0.262 − 0.124i)7-s + (−1.95 − 1.35i)8-s + (1.42 − 0.477i)9-s + (−1.23 + 2.60i)10-s + (−0.398 + 1.19i)11-s + (3.55 + 0.875i)12-s + (−0.576 + 0.816i)13-s + (−0.512 + 0.126i)14-s + (0.201 − 2.49i)15-s + (1.40 + 1.45i)16-s + (−0.556 − 1.17i)17-s + ⋯

Functional equation

Λ(s)=(169s/2 ΓC(s) L(s)=((0.369+0.929i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.369 + 0.929i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(169s/2 ΓC(s+1/2) L(s)=((0.369+0.929i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.369 + 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 169169    =    13213^{2}
Sign: 0.369+0.929i0.369 + 0.929i
Analytic conductor: 1.349471.34947
Root analytic conductor: 1.161661.16166
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ169(10,⋅)\chi_{169} (10, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 169, ( :1/2), 0.369+0.929i)(2,\ 169,\ (\ :1/2),\ 0.369 + 0.929i)

Particular Values

L(1)L(1) ≈\approx 0.738024−0.500961i0.738024 - 0.500961i
L(12)L(\frac12) ≈\approx 0.738024−0.500961i0.738024 - 0.500961i
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+(2.08−2.94i)T 1 + (2.08 - 2.94i)T
good2 1+(2.52+0.514i)T+(1.83+0.783i)T2 1 + (2.52 + 0.514i)T + (1.83 + 0.783i)T^{2}
3 1+(−2.70+0.439i)T+(2.84−0.950i)T2 1 + (-2.70 + 0.439i)T + (2.84 - 0.950i)T^{2}
5 1+(−0.847+3.43i)T+(−4.42−2.32i)T2 1 + (-0.847 + 3.43i)T + (-4.42 - 2.32i)T^{2}
7 1+(−0.693+0.328i)T+(4.42−5.42i)T2 1 + (-0.693 + 0.328i)T + (4.42 - 5.42i)T^{2}
11 1+(1.32−3.95i)T+(−8.79−6.60i)T2 1 + (1.32 - 3.95i)T + (-8.79 - 6.60i)T^{2}
17 1+(2.29+4.83i)T+(−10.7+13.1i)T2 1 + (2.29 + 4.83i)T + (-10.7 + 13.1i)T^{2}
19 1+(−1.08−0.625i)T+(9.5+16.4i)T2 1 + (-1.08 - 0.625i)T + (9.5 + 16.4i)T^{2}
23 1+(−1.81−3.14i)T+(−11.5+19.9i)T2 1 + (-1.81 - 3.14i)T + (-11.5 + 19.9i)T^{2}
29 1+(−0.242+1.18i)T+(−26.6−11.3i)T2 1 + (-0.242 + 1.18i)T + (-26.6 - 11.3i)T^{2}
31 1+(−1.61−3.08i)T+(−17.6+25.5i)T2 1 + (-1.61 - 3.08i)T + (-17.6 + 25.5i)T^{2}
37 1+(−5.46+8.63i)T+(−15.8−33.4i)T2 1 + (-5.46 + 8.63i)T + (-15.8 - 33.4i)T^{2}
41 1+(−1.11−6.83i)T+(−38.8+12.9i)T2 1 + (-1.11 - 6.83i)T + (-38.8 + 12.9i)T^{2}
43 1+(4.94−3.12i)T+(18.4−38.8i)T2 1 + (4.94 - 3.12i)T + (18.4 - 38.8i)T^{2}
47 1+(−1.77−0.215i)T+(45.6+11.2i)T2 1 + (-1.77 - 0.215i)T + (45.6 + 11.2i)T^{2}
53 1+(−0.439+0.636i)T+(−18.7−49.5i)T2 1 + (-0.439 + 0.636i)T + (-18.7 - 49.5i)T^{2}
59 1+(3.29+3.16i)T+(2.37+58.9i)T2 1 + (3.29 + 3.16i)T + (2.37 + 58.9i)T^{2}
61 1+(−7.51+0.606i)T+(60.2−9.78i)T2 1 + (-7.51 + 0.606i)T + (60.2 - 9.78i)T^{2}
67 1+(−3.22−7.57i)T+(−46.4+48.3i)T2 1 + (-3.22 - 7.57i)T + (-46.4 + 48.3i)T^{2}
71 1+(7.05−5.75i)T+(14.2−69.5i)T2 1 + (7.05 - 5.75i)T + (14.2 - 69.5i)T^{2}
73 1+(7.64+8.62i)T+(−8.79+72.4i)T2 1 + (7.64 + 8.62i)T + (-8.79 + 72.4i)T^{2}
79 1+(−0.238+1.96i)T+(−76.7−18.9i)T2 1 + (-0.238 + 1.96i)T + (-76.7 - 18.9i)T^{2}
83 1+(−1.92+0.731i)T+(62.1−55.0i)T2 1 + (-1.92 + 0.731i)T + (62.1 - 55.0i)T^{2}
89 1+(7.96−4.59i)T+(44.5−77.0i)T2 1 + (7.96 - 4.59i)T + (44.5 - 77.0i)T^{2}
97 1+(−2.85−0.827i)T+(81.9+51.8i)T2 1 + (-2.85 - 0.827i)T + (81.9 + 51.8i)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

−12.52939881061302229783767246956, −11.50948369861076278476728389264, −9.776550951459245628348494462943, −9.452337240846122867015363356073, −8.753941108776478581823528495350, −7.81927221090528229570266208284, −7.13977966595557185307406924661, −4.65189185393078644688745264470, −2.51272053170080896808376979212, −1.51413618322468157767354325780, 2.28805317398735887105919647157, 3.14935214784112573783235061802, 6.10721768411212323166919766461, 7.20916929314645384994084711491, 8.143981709367129058840303024707, 8.694583103312550327494451012327, 9.915687513169550550466849769448, 10.44289220405895075885572777661, 11.26582800930131086140518377187, 13.31974093151996889973393178938

Graph of the ZZ-function along the critical line