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Properties

Label 2-245-245.103-c1-0-14
Degree 22
Conductor 245245
Sign 0.865−0.500i0.865 - 0.500i
Analytic cond. 1.956331.95633
Root an. cond. 1.398691.39869
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  + (−0.0698 + 0.0946i)2-s + (2.41 − 0.456i)3-s + (0.585 + 1.89i)4-s + (−1.81 + 1.29i)5-s + (−0.125 + 0.260i)6-s + (1.69 − 2.03i)7-s + (−0.442 − 0.154i)8-s + (2.82 − 1.10i)9-s + (0.00412 − 0.263i)10-s + (−0.120 + 0.305i)11-s + (2.28 + 4.31i)12-s + (1.21 + 0.137i)13-s + (0.0737 + 0.302i)14-s + (−3.79 + 3.96i)15-s + (−3.23 + 2.20i)16-s + (0.851 + 0.0318i)17-s + ⋯
L(s)  = 1  + (−0.0494 + 0.0669i)2-s + (1.39 − 0.263i)3-s + (0.292 + 0.948i)4-s + (−0.813 + 0.581i)5-s + (−0.0512 + 0.106i)6-s + (0.640 − 0.767i)7-s + (−0.156 − 0.0547i)8-s + (0.942 − 0.369i)9-s + (0.00130 − 0.0831i)10-s + (−0.0361 + 0.0922i)11-s + (0.658 + 1.24i)12-s + (0.337 + 0.0380i)13-s + (0.0197 + 0.0808i)14-s + (−0.981 + 1.02i)15-s + (−0.809 + 0.551i)16-s + (0.206 + 0.00773i)17-s + ⋯

Functional equation

Λ(s)=(245s/2 ΓC(s) L(s)=((0.865−0.500i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.865 - 0.500i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(245s/2 ΓC(s+1/2) L(s)=((0.865−0.500i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.865 - 0.500i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 245245    =    5⋅725 \cdot 7^{2}
Sign: 0.865−0.500i0.865 - 0.500i
Analytic conductor: 1.956331.95633
Root analytic conductor: 1.398691.39869
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ245(103,⋅)\chi_{245} (103, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 245, ( :1/2), 0.865−0.500i)(2,\ 245,\ (\ :1/2),\ 0.865 - 0.500i)

Particular Values

L(1)L(1) ≈\approx 1.71676+0.460264i1.71676 + 0.460264i
L(12)L(\frac12) ≈\approx 1.71676+0.460264i1.71676 + 0.460264i
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)
bad5 1+(1.81−1.29i)T 1 + (1.81 - 1.29i)T
7 1+(−1.69+2.03i)T 1 + (-1.69 + 2.03i)T
good2 1+(0.0698−0.0946i)T+(−0.589−1.91i)T2 1 + (0.0698 - 0.0946i)T + (-0.589 - 1.91i)T^{2}
3 1+(−2.41+0.456i)T+(2.79−1.09i)T2 1 + (-2.41 + 0.456i)T + (2.79 - 1.09i)T^{2}
11 1+(0.120−0.305i)T+(−8.06−7.48i)T2 1 + (0.120 - 0.305i)T + (-8.06 - 7.48i)T^{2}
13 1+(−1.21−0.137i)T+(12.6+2.89i)T2 1 + (-1.21 - 0.137i)T + (12.6 + 2.89i)T^{2}
17 1+(−0.851−0.0318i)T+(16.9+1.27i)T2 1 + (-0.851 - 0.0318i)T + (16.9 + 1.27i)T^{2}
19 1+(1.69−2.92i)T+(−9.5−16.4i)T2 1 + (1.69 - 2.92i)T + (-9.5 - 16.4i)T^{2}
23 1+(0.287+7.67i)T+(−22.9+1.71i)T2 1 + (0.287 + 7.67i)T + (-22.9 + 1.71i)T^{2}
29 1+(8.41−1.92i)T+(26.1−12.5i)T2 1 + (8.41 - 1.92i)T + (26.1 - 12.5i)T^{2}
31 1+(−8.13+4.69i)T+(15.5−26.8i)T2 1 + (-8.13 + 4.69i)T + (15.5 - 26.8i)T^{2}
37 1+(2.69−1.42i)T+(20.8−30.5i)T2 1 + (2.69 - 1.42i)T + (20.8 - 30.5i)T^{2}
41 1+(3.82+7.93i)T+(−25.5+32.0i)T2 1 + (3.82 + 7.93i)T + (-25.5 + 32.0i)T^{2}
43 1+(−0.640−1.83i)T+(−33.6+26.8i)T2 1 + (-0.640 - 1.83i)T + (-33.6 + 26.8i)T^{2}
47 1+(−3.54−2.61i)T+(13.8+44.9i)T2 1 + (-3.54 - 2.61i)T + (13.8 + 44.9i)T^{2}
53 1+(4.73+2.50i)T+(29.8+43.7i)T2 1 + (4.73 + 2.50i)T + (29.8 + 43.7i)T^{2}
59 1+(−0.633−8.45i)T+(−58.3+8.79i)T2 1 + (-0.633 - 8.45i)T + (-58.3 + 8.79i)T^{2}
61 1+(−2.31+7.48i)T+(−50.4−34.3i)T2 1 + (-2.31 + 7.48i)T + (-50.4 - 34.3i)T^{2}
67 1+(−1.73−0.465i)T+(58.0+33.5i)T2 1 + (-1.73 - 0.465i)T + (58.0 + 33.5i)T^{2}
71 1+(−0.413+1.81i)T+(−63.9−30.8i)T2 1 + (-0.413 + 1.81i)T + (-63.9 - 30.8i)T^{2}
73 1+(4.30−3.17i)T+(21.5−69.7i)T2 1 + (4.30 - 3.17i)T + (21.5 - 69.7i)T^{2}
79 1+(6.73+3.88i)T+(39.5+68.4i)T2 1 + (6.73 + 3.88i)T + (39.5 + 68.4i)T^{2}
83 1+(−1.40−12.4i)T+(−80.9+18.4i)T2 1 + (-1.40 - 12.4i)T + (-80.9 + 18.4i)T^{2}
89 1+(0.319+0.813i)T+(−65.2+60.5i)T2 1 + (0.319 + 0.813i)T + (-65.2 + 60.5i)T^{2}
97 1+(−10.5−10.5i)T+97iT2 1 + (-10.5 - 10.5i)T + 97iT^{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

−12.24994780042960179104167961404, −11.26368816842480631389669857685, −10.33531437068492537242624156246, −8.802414582210781271080934441933, −8.081513317484871648015406053586, −7.58319285641354755555142305022, −6.65320384260970198183746513968, −4.23388391847058664889562716552, −3.50272208948586900676121401810, −2.29519143837248401237419283648, 1.73576942873223434955064920236, 3.18680763071696161993581195002, 4.58876908895335317122637210429, 5.69825606746902540400699513711, 7.35361044064832615348136117732, 8.347460931225057682157438508021, 9.001351574925219031162145328839, 9.820137370590556414725231262937, 11.17413262940080206726006120697, 11.79597095819413741591049647729

Graph of the ZZ-function along the critical line