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Properties

Label 2-13e2-169.127-c1-0-1
Degree 22
Conductor 169169
Sign 0.781−0.624i0.781 - 0.624i
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  + (−2.41 + 0.0974i)2-s + (−0.437 − 1.51i)3-s + (3.84 − 0.310i)4-s + (−2.71 + 1.03i)5-s + (1.20 + 3.60i)6-s + (−0.958 + 2.25i)7-s + (−4.46 + 0.541i)8-s + (0.446 − 0.282i)9-s + (6.47 − 2.75i)10-s + (0.842 − 1.33i)11-s + (−2.15 − 5.66i)12-s + (2.05 + 2.96i)13-s + (2.09 − 5.53i)14-s + (2.74 + 3.65i)15-s + (3.12 − 0.507i)16-s + (1.77 + 0.757i)17-s + ⋯
L(s)  = 1  + (−1.70 + 0.0689i)2-s + (−0.252 − 0.871i)3-s + (1.92 − 0.155i)4-s + (−1.21 + 0.460i)5-s + (0.491 + 1.47i)6-s + (−0.362 + 0.850i)7-s + (−1.57 + 0.191i)8-s + (0.148 − 0.0941i)9-s + (2.04 − 0.871i)10-s + (0.253 − 0.401i)11-s + (−0.620 − 1.63i)12-s + (0.570 + 0.821i)13-s + (0.561 − 1.47i)14-s + (0.708 + 0.943i)15-s + (0.780 − 0.126i)16-s + (0.431 + 0.183i)17-s + ⋯

Functional equation

Λ(s)=(169s/2 ΓC(s) L(s)=((0.781−0.624i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.781 - 0.624i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(169s/2 ΓC(s+1/2) L(s)=((0.781−0.624i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.781 - 0.624i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 169169    =    13213^{2}
Sign: 0.781−0.624i0.781 - 0.624i
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), 0.781−0.624i)(2,\ 169,\ (\ :1/2),\ 0.781 - 0.624i)

Particular Values

L(1)L(1) ≈\approx 0.349518+0.122549i0.349518 + 0.122549i
L(12)L(\frac12) ≈\approx 0.349518+0.122549i0.349518 + 0.122549i
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+(−2.05−2.96i)T 1 + (-2.05 - 2.96i)T
good2 1+(2.41−0.0974i)T+(1.99−0.160i)T2 1 + (2.41 - 0.0974i)T + (1.99 - 0.160i)T^{2}
3 1+(0.437+1.51i)T+(−2.53+1.60i)T2 1 + (0.437 + 1.51i)T + (-2.53 + 1.60i)T^{2}
5 1+(2.71−1.03i)T+(3.74−3.31i)T2 1 + (2.71 - 1.03i)T + (3.74 - 3.31i)T^{2}
7 1+(0.958−2.25i)T+(−4.84−5.04i)T2 1 + (0.958 - 2.25i)T + (-4.84 - 5.04i)T^{2}
11 1+(−0.842+1.33i)T+(−4.71−9.93i)T2 1 + (-0.842 + 1.33i)T + (-4.71 - 9.93i)T^{2}
17 1+(−1.77−0.757i)T+(11.7+12.2i)T2 1 + (-1.77 - 0.757i)T + (11.7 + 12.2i)T^{2}
19 1+(−6.36−3.67i)T+(9.5+16.4i)T2 1 + (-6.36 - 3.67i)T + (9.5 + 16.4i)T^{2}
23 1+(−2.38−4.13i)T+(−11.5+19.9i)T2 1 + (-2.38 - 4.13i)T + (-11.5 + 19.9i)T^{2}
29 1+(−0.291−7.23i)T+(−28.9+2.33i)T2 1 + (-0.291 - 7.23i)T + (-28.9 + 2.33i)T^{2}
31 1+(−3.48+3.93i)T+(−3.73−30.7i)T2 1 + (-3.48 + 3.93i)T + (-3.73 - 30.7i)T^{2}
37 1+(9.28+1.89i)T+(34.0+14.5i)T2 1 + (9.28 + 1.89i)T + (34.0 + 14.5i)T^{2}
41 1+(1.54−0.447i)T+(34.6−21.9i)T2 1 + (1.54 - 0.447i)T + (34.6 - 21.9i)T^{2}
43 1+(−0.644−3.15i)T+(−39.5+16.8i)T2 1 + (-0.644 - 3.15i)T + (-39.5 + 16.8i)T^{2}
47 1+(0.644+0.444i)T+(16.6+43.9i)T2 1 + (0.644 + 0.444i)T + (16.6 + 43.9i)T^{2}
53 1+(−1.39−11.4i)T+(−51.4+12.6i)T2 1 + (-1.39 - 11.4i)T + (-51.4 + 12.6i)T^{2}
59 1+(0.344−2.12i)T+(−55.9−18.6i)T2 1 + (0.344 - 2.12i)T + (-55.9 - 18.6i)T^{2}
61 1+(7.39+5.55i)T+(16.9+58.5i)T2 1 + (7.39 + 5.55i)T + (16.9 + 58.5i)T^{2}
67 1+(0.146−1.81i)T+(−66.1−10.7i)T2 1 + (0.146 - 1.81i)T + (-66.1 - 10.7i)T^{2}
71 1+(−8.24−7.91i)T+(2.85+70.9i)T2 1 + (-8.24 - 7.91i)T + (2.85 + 70.9i)T^{2}
73 1+(3.61+6.88i)T+(−41.4+60.0i)T2 1 + (3.61 + 6.88i)T + (-41.4 + 60.0i)T^{2}
79 1+(−3.01+4.36i)T+(−28.0−73.8i)T2 1 + (-3.01 + 4.36i)T + (-28.0 - 73.8i)T^{2}
83 1+(−0.682−2.76i)T+(−73.4+38.5i)T2 1 + (-0.682 - 2.76i)T + (-73.4 + 38.5i)T^{2}
89 1+(−10.7+6.18i)T+(44.5−77.0i)T2 1 + (-10.7 + 6.18i)T + (44.5 - 77.0i)T^{2}
97 1+(10.8−8.83i)T+(19.4−95.0i)T2 1 + (10.8 - 8.83i)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

−12.23940383048358006306537849463, −11.79073495293906981508603973367, −10.95126186009872705253737149509, −9.637842160647554949716617935933, −8.746063285415544060199633057213, −7.68816481187188807312820712944, −7.09011253873719660935358268936, −6.02771091501194485747648972905, −3.38543480092005756570974743923, −1.37423685818158815826927494890, 0.71490873714128110135936236676, 3.50354183801910220170761603926, 4.84924907516371549306694881196, 6.94549624060574498920724531652, 7.72621608912310840406925122175, 8.678407938003848293348835979369, 9.794646601151361824396057739277, 10.39667222856493729649801979041, 11.27622513998618652106785762210, 12.14221014066028751488585518539

Graph of the ZZ-function along the critical line