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Properties

Label 2-357-119.10-c1-0-0
Degree 22
Conductor 357357
Sign −0.990+0.139i-0.990 + 0.139i
Analytic cond. 2.850652.85065
Root an. cond. 1.688381.68838
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  + (0.0808 − 0.0106i)2-s + (0.442 + 0.896i)3-s + (−1.92 + 0.515i)4-s + (0.199 − 0.174i)5-s + (0.0453 + 0.0678i)6-s + (−2.58 − 0.554i)7-s + (−0.300 + 0.124i)8-s + (−0.608 + 0.793i)9-s + (0.0142 − 0.0162i)10-s + (−6.03 + 0.395i)11-s + (−1.31 − 1.49i)12-s + (−1.97 − 1.97i)13-s + (−0.215 − 0.0172i)14-s + (0.245 + 0.101i)15-s + (3.42 − 1.98i)16-s + (3.81 + 1.55i)17-s + ⋯
L(s)  = 1  + (0.0571 − 0.00752i)2-s + (0.255 + 0.517i)3-s + (−0.962 + 0.257i)4-s + (0.0891 − 0.0781i)5-s + (0.0184 + 0.0276i)6-s + (−0.977 − 0.209i)7-s + (−0.106 + 0.0440i)8-s + (−0.202 + 0.264i)9-s + (0.00450 − 0.00514i)10-s + (−1.81 + 0.119i)11-s + (−0.379 − 0.432i)12-s + (−0.547 − 0.547i)13-s + (−0.0574 − 0.00461i)14-s + (0.0632 + 0.0262i)15-s + (0.857 − 0.495i)16-s + (0.926 + 0.376i)17-s + ⋯

Functional equation

Λ(s)=(357s/2 ΓC(s) L(s)=((−0.990+0.139i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 357 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.990 + 0.139i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(357s/2 ΓC(s+1/2) L(s)=((−0.990+0.139i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 357 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.990 + 0.139i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 357357    =    3⋅7⋅173 \cdot 7 \cdot 17
Sign: −0.990+0.139i-0.990 + 0.139i
Analytic conductor: 2.850652.85065
Root analytic conductor: 1.688381.68838
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ357(10,⋅)\chi_{357} (10, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 357, ( :1/2), −0.990+0.139i)(2,\ 357,\ (\ :1/2),\ -0.990 + 0.139i)

Particular Values

L(1)L(1) ≈\approx 0.0121955−0.174259i0.0121955 - 0.174259i
L(12)L(\frac12) ≈\approx 0.0121955−0.174259i0.0121955 - 0.174259i
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)
bad3 1+(−0.442−0.896i)T 1 + (-0.442 - 0.896i)T
7 1+(2.58+0.554i)T 1 + (2.58 + 0.554i)T
17 1+(−3.81−1.55i)T 1 + (-3.81 - 1.55i)T
good2 1+(−0.0808+0.0106i)T+(1.93−0.517i)T2 1 + (-0.0808 + 0.0106i)T + (1.93 - 0.517i)T^{2}
5 1+(−0.199+0.174i)T+(0.652−4.95i)T2 1 + (-0.199 + 0.174i)T + (0.652 - 4.95i)T^{2}
11 1+(6.03−0.395i)T+(10.9−1.43i)T2 1 + (6.03 - 0.395i)T + (10.9 - 1.43i)T^{2}
13 1+(1.97+1.97i)T+13iT2 1 + (1.97 + 1.97i)T + 13iT^{2}
19 1+(−0.124−0.943i)T+(−18.3+4.91i)T2 1 + (-0.124 - 0.943i)T + (-18.3 + 4.91i)T^{2}
23 1+(5.23+2.58i)T+(14.0+18.2i)T2 1 + (5.23 + 2.58i)T + (14.0 + 18.2i)T^{2}
29 1+(0.808+4.06i)T+(−26.7+11.0i)T2 1 + (0.808 + 4.06i)T + (-26.7 + 11.0i)T^{2}
31 1+(6.53−3.22i)T+(18.8−24.5i)T2 1 + (6.53 - 3.22i)T + (18.8 - 24.5i)T^{2}
37 1+(0.141−2.16i)T+(−36.6−4.82i)T2 1 + (0.141 - 2.16i)T + (-36.6 - 4.82i)T^{2}
41 1+(0.435−2.19i)T+(−37.8−15.6i)T2 1 + (0.435 - 2.19i)T + (-37.8 - 15.6i)T^{2}
43 1+(−4.25−10.2i)T+(−30.4+30.4i)T2 1 + (-4.25 - 10.2i)T + (-30.4 + 30.4i)T^{2}
47 1+(−1.94+7.25i)T+(−40.7−23.5i)T2 1 + (-1.94 + 7.25i)T + (-40.7 - 23.5i)T^{2}
53 1+(1.17+1.53i)T+(−13.7+51.1i)T2 1 + (1.17 + 1.53i)T + (-13.7 + 51.1i)T^{2}
59 1+(10.4+1.37i)T+(56.9+15.2i)T2 1 + (10.4 + 1.37i)T + (56.9 + 15.2i)T^{2}
61 1+(−2.64−7.78i)T+(−48.3+37.1i)T2 1 + (-2.64 - 7.78i)T + (-48.3 + 37.1i)T^{2}
67 1+(−2.89−1.67i)T+(33.5+58.0i)T2 1 + (-2.89 - 1.67i)T + (33.5 + 58.0i)T^{2}
71 1+(−4.95−3.31i)T+(27.1+65.5i)T2 1 + (-4.95 - 3.31i)T + (27.1 + 65.5i)T^{2}
73 1+(−3.65−1.24i)T+(57.9+44.4i)T2 1 + (-3.65 - 1.24i)T + (57.9 + 44.4i)T^{2}
79 1+(−7.03+14.2i)T+(−48.0−62.6i)T2 1 + (-7.03 + 14.2i)T + (-48.0 - 62.6i)T^{2}
83 1+(−1.65+3.98i)T+(−58.6−58.6i)T2 1 + (-1.65 + 3.98i)T + (-58.6 - 58.6i)T^{2}
89 1+(1.34+0.361i)T+(77.0+44.5i)T2 1 + (1.34 + 0.361i)T + (77.0 + 44.5i)T^{2}
97 1+(14.5−2.89i)T+(89.6−37.1i)T2 1 + (14.5 - 2.89i)T + (89.6 - 37.1i)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

−12.24445372082496032668541313554, −10.60470159766498781701498514512, −10.01740005687417429403932874731, −9.385417096954071364845837351229, −8.138414291369994996349662761453, −7.59325829285405432117082834546, −5.80474355689114893062952419639, −5.04788925287027850131806559876, −3.78729225339925751646431417285, −2.82542338835172741213081271505, 0.10900903058998658621620641539, 2.42911959041092733543775235492, 3.69186013476904055465777594556, 5.19614105694680673109270951122, 5.93451495775572580556076371039, 7.32109366488140661305101676588, 8.115599684678223190815873101648, 9.263049232152314234574022663607, 9.869357113939168449094645343407, 10.78368411602991671182188130092

Graph of the ZZ-function along the critical line