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Properties

Label 2-13e2-169.127-c1-0-12
Degree 22
Conductor 169169
Sign 0.866+0.499i0.866 + 0.499i
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  + (2.57 − 0.103i)2-s + (−0.381 − 1.31i)3-s + (4.64 − 0.374i)4-s + (−3.32 + 1.26i)5-s + (−1.12 − 3.35i)6-s + (0.633 − 1.48i)7-s + (6.81 − 0.827i)8-s + (0.946 − 0.598i)9-s + (−8.44 + 3.59i)10-s + (−2.92 + 4.61i)11-s + (−2.26 − 5.97i)12-s + (0.0292 + 3.60i)13-s + (1.47 − 3.90i)14-s + (2.92 + 3.89i)15-s + (8.28 − 1.34i)16-s + (0.607 + 0.258i)17-s + ⋯
L(s)  = 1  + (1.82 − 0.0734i)2-s + (−0.220 − 0.760i)3-s + (2.32 − 0.187i)4-s + (−1.48 + 0.563i)5-s + (−0.457 − 1.37i)6-s + (0.239 − 0.562i)7-s + (2.40 − 0.292i)8-s + (0.315 − 0.199i)9-s + (−2.66 + 1.13i)10-s + (−0.880 + 1.39i)11-s + (−0.654 − 1.72i)12-s + (0.00810 + 0.999i)13-s + (0.395 − 1.04i)14-s + (0.756 + 1.00i)15-s + (2.07 − 0.336i)16-s + (0.147 + 0.0628i)17-s + ⋯

Functional equation

Λ(s)=(169s/2 ΓC(s) L(s)=((0.866+0.499i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.866 + 0.499i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(169s/2 ΓC(s+1/2) L(s)=((0.866+0.499i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.866 + 0.499i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 169169    =    13213^{2}
Sign: 0.866+0.499i0.866 + 0.499i
Analytic conductor: 1.349471.34947
Root analytic conductor: 1.161661.16166
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ169(127,⋅)\chi_{169} (127, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 169, ( :1/2), 0.866+0.499i)(2,\ 169,\ (\ :1/2),\ 0.866 + 0.499i)

Particular Values

L(1)L(1) ≈\approx 2.29606−0.614350i2.29606 - 0.614350i
L(12)L(\frac12) ≈\approx 2.29606−0.614350i2.29606 - 0.614350i
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+(−0.0292−3.60i)T 1 + (-0.0292 - 3.60i)T
good2 1+(−2.57+0.103i)T+(1.99−0.160i)T2 1 + (-2.57 + 0.103i)T + (1.99 - 0.160i)T^{2}
3 1+(0.381+1.31i)T+(−2.53+1.60i)T2 1 + (0.381 + 1.31i)T + (-2.53 + 1.60i)T^{2}
5 1+(3.32−1.26i)T+(3.74−3.31i)T2 1 + (3.32 - 1.26i)T + (3.74 - 3.31i)T^{2}
7 1+(−0.633+1.48i)T+(−4.84−5.04i)T2 1 + (-0.633 + 1.48i)T + (-4.84 - 5.04i)T^{2}
11 1+(2.92−4.61i)T+(−4.71−9.93i)T2 1 + (2.92 - 4.61i)T + (-4.71 - 9.93i)T^{2}
17 1+(−0.607−0.258i)T+(11.7+12.2i)T2 1 + (-0.607 - 0.258i)T + (11.7 + 12.2i)T^{2}
19 1+(3.26+1.88i)T+(9.5+16.4i)T2 1 + (3.26 + 1.88i)T + (9.5 + 16.4i)T^{2}
23 1+(2.06+3.58i)T+(−11.5+19.9i)T2 1 + (2.06 + 3.58i)T + (-11.5 + 19.9i)T^{2}
29 1+(0.0150+0.374i)T+(−28.9+2.33i)T2 1 + (0.0150 + 0.374i)T + (-28.9 + 2.33i)T^{2}
31 1+(−4.01+4.52i)T+(−3.73−30.7i)T2 1 + (-4.01 + 4.52i)T + (-3.73 - 30.7i)T^{2}
37 1+(−8.68−1.77i)T+(34.0+14.5i)T2 1 + (-8.68 - 1.77i)T + (34.0 + 14.5i)T^{2}
41 1+(6.39−1.85i)T+(34.6−21.9i)T2 1 + (6.39 - 1.85i)T + (34.6 - 21.9i)T^{2}
43 1+(1.58+7.75i)T+(−39.5+16.8i)T2 1 + (1.58 + 7.75i)T + (-39.5 + 16.8i)T^{2}
47 1+(−7.99−5.51i)T+(16.6+43.9i)T2 1 + (-7.99 - 5.51i)T + (16.6 + 43.9i)T^{2}
53 1+(0.387+3.18i)T+(−51.4+12.6i)T2 1 + (0.387 + 3.18i)T + (-51.4 + 12.6i)T^{2}
59 1+(−0.599+3.68i)T+(−55.9−18.6i)T2 1 + (-0.599 + 3.68i)T + (-55.9 - 18.6i)T^{2}
61 1+(3.97+2.98i)T+(16.9+58.5i)T2 1 + (3.97 + 2.98i)T + (16.9 + 58.5i)T^{2}
67 1+(0.700−8.67i)T+(−66.1−10.7i)T2 1 + (0.700 - 8.67i)T + (-66.1 - 10.7i)T^{2}
71 1+(−0.998−0.959i)T+(2.85+70.9i)T2 1 + (-0.998 - 0.959i)T + (2.85 + 70.9i)T^{2}
73 1+(0.592+1.12i)T+(−41.4+60.0i)T2 1 + (0.592 + 1.12i)T + (-41.4 + 60.0i)T^{2}
79 1+(2.22−3.22i)T+(−28.0−73.8i)T2 1 + (2.22 - 3.22i)T + (-28.0 - 73.8i)T^{2}
83 1+(1.29+5.24i)T+(−73.4+38.5i)T2 1 + (1.29 + 5.24i)T + (-73.4 + 38.5i)T^{2}
89 1+(−2.98+1.72i)T+(44.5−77.0i)T2 1 + (-2.98 + 1.72i)T + (44.5 - 77.0i)T^{2}
97 1+(10.3−8.45i)T+(19.4−95.0i)T2 1 + (10.3 - 8.45i)T + (19.4 - 95.0i)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

−12.64640265602941064829560031833, −12.01635330059612169693068059651, −11.31750203883927704914908192749, −10.29030827759539380149917148403, −7.79874151829973167106590401954, −7.15109176609073055423611246800, −6.43600089209193469749301192812, −4.54933281947871801293271547763, −4.07672747787705185962972979794, −2.35291043309733468343661329194, 3.10954806284263771463929008899, 4.08277376286960215430961535525, 5.06005798576170379983312532180, 5.83006024328164707892476110919, 7.60024283378501980898845480484, 8.397000478131389877952744805818, 10.48701012263283796500286252108, 11.22847234957584780590050571639, 12.04054084106673080322715294566, 12.84547465457740212428683348226

Graph of the ZZ-function along the critical line