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Properties

Label 2-13e2-169.10-c1-0-1
Degree 22
Conductor 169169
Sign 0.395−0.918i0.395 - 0.918i
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  + (−1.44 − 0.294i)2-s + (−2.46 + 0.401i)3-s + (0.148 + 0.0631i)4-s + (0.720 − 2.92i)5-s + (3.67 + 0.148i)6-s + (−2.39 + 1.13i)7-s + (2.22 + 1.53i)8-s + (3.09 − 1.03i)9-s + (−1.89 + 3.99i)10-s + (−0.728 + 2.18i)11-s + (−0.391 − 0.0965i)12-s + (1.92 + 3.04i)13-s + (3.78 − 0.932i)14-s + (−0.606 + 7.51i)15-s + (−2.97 − 3.09i)16-s + (1.97 + 4.16i)17-s + ⋯
L(s)  = 1  + (−1.01 − 0.207i)2-s + (−1.42 + 0.231i)3-s + (0.0741 + 0.0315i)4-s + (0.322 − 1.30i)5-s + (1.50 + 0.0604i)6-s + (−0.904 + 0.429i)7-s + (0.786 + 0.542i)8-s + (1.03 − 0.344i)9-s + (−0.600 + 1.26i)10-s + (−0.219 + 0.657i)11-s + (−0.113 − 0.0278i)12-s + (0.534 + 0.845i)13-s + (1.01 − 0.249i)14-s + (−0.156 + 1.93i)15-s + (−0.744 − 0.774i)16-s + (0.479 + 1.01i)17-s + ⋯

Functional equation

Λ(s)=(169s/2 ΓC(s) L(s)=((0.395−0.918i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.395 - 0.918i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(169s/2 ΓC(s+1/2) L(s)=((0.395−0.918i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.395 - 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 169169    =    13213^{2}
Sign: 0.395−0.918i0.395 - 0.918i
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), 0.395−0.918i)(2,\ 169,\ (\ :1/2),\ 0.395 - 0.918i)

Particular Values

L(1)L(1) ≈\approx 0.222002+0.146195i0.222002 + 0.146195i
L(12)L(\frac12) ≈\approx 0.222002+0.146195i0.222002 + 0.146195i
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+(−1.92−3.04i)T 1 + (-1.92 - 3.04i)T
good2 1+(1.44+0.294i)T+(1.83+0.783i)T2 1 + (1.44 + 0.294i)T + (1.83 + 0.783i)T^{2}
3 1+(2.46−0.401i)T+(2.84−0.950i)T2 1 + (2.46 - 0.401i)T + (2.84 - 0.950i)T^{2}
5 1+(−0.720+2.92i)T+(−4.42−2.32i)T2 1 + (-0.720 + 2.92i)T + (-4.42 - 2.32i)T^{2}
7 1+(2.39−1.13i)T+(4.42−5.42i)T2 1 + (2.39 - 1.13i)T + (4.42 - 5.42i)T^{2}
11 1+(0.728−2.18i)T+(−8.79−6.60i)T2 1 + (0.728 - 2.18i)T + (-8.79 - 6.60i)T^{2}
17 1+(−1.97−4.16i)T+(−10.7+13.1i)T2 1 + (-1.97 - 4.16i)T + (-10.7 + 13.1i)T^{2}
19 1+(1.79+1.03i)T+(9.5+16.4i)T2 1 + (1.79 + 1.03i)T + (9.5 + 16.4i)T^{2}
23 1+(−3.57−6.18i)T+(−11.5+19.9i)T2 1 + (-3.57 - 6.18i)T + (-11.5 + 19.9i)T^{2}
29 1+(−0.932+4.56i)T+(−26.6−11.3i)T2 1 + (-0.932 + 4.56i)T + (-26.6 - 11.3i)T^{2}
31 1+(1.28+2.44i)T+(−17.6+25.5i)T2 1 + (1.28 + 2.44i)T + (-17.6 + 25.5i)T^{2}
37 1+(1.42−2.24i)T+(−15.8−33.4i)T2 1 + (1.42 - 2.24i)T + (-15.8 - 33.4i)T^{2}
41 1+(−1.11−6.88i)T+(−38.8+12.9i)T2 1 + (-1.11 - 6.88i)T + (-38.8 + 12.9i)T^{2}
43 1+(7.73−4.89i)T+(18.4−38.8i)T2 1 + (7.73 - 4.89i)T + (18.4 - 38.8i)T^{2}
47 1+(1.71+0.208i)T+(45.6+11.2i)T2 1 + (1.71 + 0.208i)T + (45.6 + 11.2i)T^{2}
53 1+(7.11−10.3i)T+(−18.7−49.5i)T2 1 + (7.11 - 10.3i)T + (-18.7 - 49.5i)T^{2}
59 1+(−8.35−8.02i)T+(2.37+58.9i)T2 1 + (-8.35 - 8.02i)T + (2.37 + 58.9i)T^{2}
61 1+(−3.47+0.280i)T+(60.2−9.78i)T2 1 + (-3.47 + 0.280i)T + (60.2 - 9.78i)T^{2}
67 1+(5.31+12.4i)T+(−46.4+48.3i)T2 1 + (5.31 + 12.4i)T + (-46.4 + 48.3i)T^{2}
71 1+(−8.65+7.06i)T+(14.2−69.5i)T2 1 + (-8.65 + 7.06i)T + (14.2 - 69.5i)T^{2}
73 1+(−0.619−0.699i)T+(−8.79+72.4i)T2 1 + (-0.619 - 0.699i)T + (-8.79 + 72.4i)T^{2}
79 1+(1.68−13.8i)T+(−76.7−18.9i)T2 1 + (1.68 - 13.8i)T + (-76.7 - 18.9i)T^{2}
83 1+(−10.5+4.00i)T+(62.1−55.0i)T2 1 + (-10.5 + 4.00i)T + (62.1 - 55.0i)T^{2}
89 1+(7.69−4.44i)T+(44.5−77.0i)T2 1 + (7.69 - 4.44i)T + (44.5 - 77.0i)T^{2}
97 1+(9.44+2.73i)T+(81.9+51.8i)T2 1 + (9.44 + 2.73i)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

−12.74934069296589368183990701928, −11.81766695358611231948847576032, −10.86564017545505473404085997164, −9.745821921027399120758646144194, −9.303596215986027229581877421268, −8.131611435298801976528116442574, −6.45873161509662724761289702308, −5.42615482202931518155753945847, −4.46888574247211617584142449066, −1.38198983670162624157366984553, 0.45743614550339639204461641449, 3.31596284240950381315531093039, 5.35915231802218136668157178878, 6.65775049977064124559100888880, 6.98269313766089760147667129572, 8.486376186678968172024393972190, 9.952635410077321135051390843237, 10.55850439891737605857699768611, 11.09986090877292658010492009873, 12.55436560071802421522821959802

Graph of the ZZ-function along the critical line