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Properties

Label 2-13e2-169.127-c1-0-13
Degree 22
Conductor 169169
Sign 0.469+0.882i0.469 + 0.882i
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  + (1.99 − 0.0804i)2-s + (−0.751 − 2.59i)3-s + (1.98 − 0.160i)4-s + (1.41 − 0.534i)5-s + (−1.70 − 5.11i)6-s + (−1.63 + 3.84i)7-s + (−0.0203 + 0.00246i)8-s + (−3.62 + 2.29i)9-s + (2.77 − 1.18i)10-s + (3.00 − 4.75i)11-s + (−1.90 − 5.02i)12-s + (−0.967 + 3.47i)13-s + (−2.95 + 7.80i)14-s + (−2.44 − 3.25i)15-s + (−3.96 + 0.644i)16-s + (6.45 + 2.74i)17-s + ⋯
L(s)  = 1  + (1.41 − 0.0568i)2-s + (−0.433 − 1.49i)3-s + (0.991 − 0.0800i)4-s + (0.630 − 0.239i)5-s + (−0.697 − 2.08i)6-s + (−0.619 + 1.45i)7-s + (−0.00719 + 0.000873i)8-s + (−1.20 + 0.765i)9-s + (0.876 − 0.373i)10-s + (0.907 − 1.43i)11-s + (−0.550 − 1.45i)12-s + (−0.268 + 0.963i)13-s + (−0.791 + 2.08i)14-s + (−0.631 − 0.840i)15-s + (−0.992 + 0.161i)16-s + (1.56 + 0.666i)17-s + ⋯

Functional equation

Λ(s)=(169s/2 ΓC(s) L(s)=((0.469+0.882i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.469 + 0.882i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(169s/2 ΓC(s+1/2) L(s)=((0.469+0.882i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.469 + 0.882i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 169169    =    13213^{2}
Sign: 0.469+0.882i0.469 + 0.882i
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), 0.469+0.882i)(2,\ 169,\ (\ :1/2),\ 0.469 + 0.882i)

Particular Values

L(1)L(1) ≈\approx 1.71236−1.02901i1.71236 - 1.02901i
L(12)L(\frac12) ≈\approx 1.71236−1.02901i1.71236 - 1.02901i
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+(0.967−3.47i)T 1 + (0.967 - 3.47i)T
good2 1+(−1.99+0.0804i)T+(1.99−0.160i)T2 1 + (-1.99 + 0.0804i)T + (1.99 - 0.160i)T^{2}
3 1+(0.751+2.59i)T+(−2.53+1.60i)T2 1 + (0.751 + 2.59i)T + (-2.53 + 1.60i)T^{2}
5 1+(−1.41+0.534i)T+(3.74−3.31i)T2 1 + (-1.41 + 0.534i)T + (3.74 - 3.31i)T^{2}
7 1+(1.63−3.84i)T+(−4.84−5.04i)T2 1 + (1.63 - 3.84i)T + (-4.84 - 5.04i)T^{2}
11 1+(−3.00+4.75i)T+(−4.71−9.93i)T2 1 + (-3.00 + 4.75i)T + (-4.71 - 9.93i)T^{2}
17 1+(−6.45−2.74i)T+(11.7+12.2i)T2 1 + (-6.45 - 2.74i)T + (11.7 + 12.2i)T^{2}
19 1+(1.84+1.06i)T+(9.5+16.4i)T2 1 + (1.84 + 1.06i)T + (9.5 + 16.4i)T^{2}
23 1+(−0.789−1.36i)T+(−11.5+19.9i)T2 1 + (-0.789 - 1.36i)T + (-11.5 + 19.9i)T^{2}
29 1+(0.179+4.46i)T+(−28.9+2.33i)T2 1 + (0.179 + 4.46i)T + (-28.9 + 2.33i)T^{2}
31 1+(2.69−3.04i)T+(−3.73−30.7i)T2 1 + (2.69 - 3.04i)T + (-3.73 - 30.7i)T^{2}
37 1+(−3.77−0.771i)T+(34.0+14.5i)T2 1 + (-3.77 - 0.771i)T + (34.0 + 14.5i)T^{2}
41 1+(4.49−1.30i)T+(34.6−21.9i)T2 1 + (4.49 - 1.30i)T + (34.6 - 21.9i)T^{2}
43 1+(0.625+3.06i)T+(−39.5+16.8i)T2 1 + (0.625 + 3.06i)T + (-39.5 + 16.8i)T^{2}
47 1+(6.46+4.46i)T+(16.6+43.9i)T2 1 + (6.46 + 4.46i)T + (16.6 + 43.9i)T^{2}
53 1+(0.932+7.68i)T+(−51.4+12.6i)T2 1 + (0.932 + 7.68i)T + (-51.4 + 12.6i)T^{2}
59 1+(0.475−2.92i)T+(−55.9−18.6i)T2 1 + (0.475 - 2.92i)T + (-55.9 - 18.6i)T^{2}
61 1+(−2.12−1.59i)T+(16.9+58.5i)T2 1 + (-2.12 - 1.59i)T + (16.9 + 58.5i)T^{2}
67 1+(−0.307+3.81i)T+(−66.1−10.7i)T2 1 + (-0.307 + 3.81i)T + (-66.1 - 10.7i)T^{2}
71 1+(0.130+0.125i)T+(2.85+70.9i)T2 1 + (0.130 + 0.125i)T + (2.85 + 70.9i)T^{2}
73 1+(−3.18−6.06i)T+(−41.4+60.0i)T2 1 + (-3.18 - 6.06i)T + (-41.4 + 60.0i)T^{2}
79 1+(−5.93+8.60i)T+(−28.0−73.8i)T2 1 + (-5.93 + 8.60i)T + (-28.0 - 73.8i)T^{2}
83 1+(1.37+5.57i)T+(−73.4+38.5i)T2 1 + (1.37 + 5.57i)T + (-73.4 + 38.5i)T^{2}
89 1+(6.07−3.50i)T+(44.5−77.0i)T2 1 + (6.07 - 3.50i)T + (44.5 - 77.0i)T^{2}
97 1+(7.22−5.89i)T+(19.4−95.0i)T2 1 + (7.22 - 5.89i)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

−12.72652660454853744500189203057, −11.89312962740772014836419901823, −11.55772349611556794897175467011, −9.429731256917724867044323322804, −8.426965496619208430276695974318, −6.66802836260685388606747661991, −6.00272546239387744855730302087, −5.47943548666149444024452923773, −3.37399602224962861708691158748, −1.92634983281216017801913512478, 3.28483792683942045924106736024, 4.17120325450993884921021969867, 5.04494784967222355613684942589, 6.14929801836633023127437344918, 7.28684957807178696359444977514, 9.679441812897431866515373034059, 9.890244414111634039605314220123, 10.91174770548741446243362060586, 12.18049988095950996930461552333, 12.99209356429649658438141146108

Graph of the ZZ-function along the critical line