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Properties

Label 2-357-119.10-c1-0-22
Degree 22
Conductor 357357
Sign 0.485+0.874i0.485 + 0.874i
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  + (2.17 − 0.286i)2-s + (−0.442 − 0.896i)3-s + (2.72 − 0.729i)4-s + (1.07 − 0.945i)5-s + (−1.21 − 1.82i)6-s + (0.241 − 2.63i)7-s + (1.65 − 0.686i)8-s + (−0.608 + 0.793i)9-s + (2.07 − 2.36i)10-s + (−4.77 + 0.312i)11-s + (−1.85 − 2.11i)12-s + (4.50 + 4.50i)13-s + (−0.230 − 5.80i)14-s + (−1.32 − 0.548i)15-s + (−1.47 + 0.849i)16-s + (4.08 + 0.569i)17-s + ⋯
L(s)  = 1  + (1.53 − 0.202i)2-s + (−0.255 − 0.517i)3-s + (1.36 − 0.364i)4-s + (0.482 − 0.422i)5-s + (−0.497 − 0.745i)6-s + (0.0911 − 0.995i)7-s + (0.585 − 0.242i)8-s + (−0.202 + 0.264i)9-s + (0.656 − 0.748i)10-s + (−1.43 + 0.0943i)11-s + (−0.536 − 0.611i)12-s + (1.24 + 1.24i)13-s + (−0.0615 − 1.55i)14-s + (−0.342 − 0.141i)15-s + (−0.367 + 0.212i)16-s + (0.990 + 0.138i)17-s + ⋯

Functional equation

Λ(s)=(357s/2 ΓC(s) L(s)=((0.485+0.874i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 357 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.485 + 0.874i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(357s/2 ΓC(s+1/2) L(s)=((0.485+0.874i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 357 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.485 + 0.874i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 357357    =    3⋅7⋅173 \cdot 7 \cdot 17
Sign: 0.485+0.874i0.485 + 0.874i
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), 0.485+0.874i)(2,\ 357,\ (\ :1/2),\ 0.485 + 0.874i)

Particular Values

L(1)L(1) ≈\approx 2.43327−1.43129i2.43327 - 1.43129i
L(12)L(\frac12) ≈\approx 2.43327−1.43129i2.43327 - 1.43129i
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+(−0.241+2.63i)T 1 + (-0.241 + 2.63i)T
17 1+(−4.08−0.569i)T 1 + (-4.08 - 0.569i)T
good2 1+(−2.17+0.286i)T+(1.93−0.517i)T2 1 + (-2.17 + 0.286i)T + (1.93 - 0.517i)T^{2}
5 1+(−1.07+0.945i)T+(0.652−4.95i)T2 1 + (-1.07 + 0.945i)T + (0.652 - 4.95i)T^{2}
11 1+(4.77−0.312i)T+(10.9−1.43i)T2 1 + (4.77 - 0.312i)T + (10.9 - 1.43i)T^{2}
13 1+(−4.50−4.50i)T+13iT2 1 + (-4.50 - 4.50i)T + 13iT^{2}
19 1+(0.367+2.79i)T+(−18.3+4.91i)T2 1 + (0.367 + 2.79i)T + (-18.3 + 4.91i)T^{2}
23 1+(−1.14−0.563i)T+(14.0+18.2i)T2 1 + (-1.14 - 0.563i)T + (14.0 + 18.2i)T^{2}
29 1+(−0.234−1.17i)T+(−26.7+11.0i)T2 1 + (-0.234 - 1.17i)T + (-26.7 + 11.0i)T^{2}
31 1+(−1.66+0.819i)T+(18.8−24.5i)T2 1 + (-1.66 + 0.819i)T + (18.8 - 24.5i)T^{2}
37 1+(−0.312+4.76i)T+(−36.6−4.82i)T2 1 + (-0.312 + 4.76i)T + (-36.6 - 4.82i)T^{2}
41 1+(2.29−11.5i)T+(−37.8−15.6i)T2 1 + (2.29 - 11.5i)T + (-37.8 - 15.6i)T^{2}
43 1+(−2.19−5.28i)T+(−30.4+30.4i)T2 1 + (-2.19 - 5.28i)T + (-30.4 + 30.4i)T^{2}
47 1+(−1.67+6.26i)T+(−40.7−23.5i)T2 1 + (-1.67 + 6.26i)T + (-40.7 - 23.5i)T^{2}
53 1+(1.55+2.02i)T+(−13.7+51.1i)T2 1 + (1.55 + 2.02i)T + (-13.7 + 51.1i)T^{2}
59 1+(10.5+1.39i)T+(56.9+15.2i)T2 1 + (10.5 + 1.39i)T + (56.9 + 15.2i)T^{2}
61 1+(3.94+11.6i)T+(−48.3+37.1i)T2 1 + (3.94 + 11.6i)T + (-48.3 + 37.1i)T^{2}
67 1+(−6.44−3.71i)T+(33.5+58.0i)T2 1 + (-6.44 - 3.71i)T + (33.5 + 58.0i)T^{2}
71 1+(5.14+3.43i)T+(27.1+65.5i)T2 1 + (5.14 + 3.43i)T + (27.1 + 65.5i)T^{2}
73 1+(−1.15−0.391i)T+(57.9+44.4i)T2 1 + (-1.15 - 0.391i)T + (57.9 + 44.4i)T^{2}
79 1+(7.56−15.3i)T+(−48.0−62.6i)T2 1 + (7.56 - 15.3i)T + (-48.0 - 62.6i)T^{2}
83 1+(2.46−5.95i)T+(−58.6−58.6i)T2 1 + (2.46 - 5.95i)T + (-58.6 - 58.6i)T^{2}
89 1+(8.90+2.38i)T+(77.0+44.5i)T2 1 + (8.90 + 2.38i)T + (77.0 + 44.5i)T^{2}
97 1+(−6.87+1.36i)T+(89.6−37.1i)T2 1 + (-6.87 + 1.36i)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

−11.34790405407015850302004896708, −10.93829619270856144459701492620, −9.661189167891264672637876721681, −8.282977904086480427007762141056, −7.14334033590226109017546914959, −6.19065737470571533828261687409, −5.27470030394272585284851769418, −4.39485677389778013858927772862, −3.14304744763352530651151024428, −1.59894377841986514548915099017, 2.64605956219805954421296011702, 3.42523423202066914092145817897, 4.90528816056389194764517914427, 5.79981010581113487744293266483, 6.00286584435868646686613379094, 7.67541218233332381768427426570, 8.750172638196464669529319330837, 10.19973055803799502408752638088, 10.73094170953463970693257187243, 11.91545481809954320780700069707

Graph of the ZZ-function along the critical line