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Properties

Label 2-350-175.108-c1-0-4
Degree 22
Conductor 350350
Sign 0.522−0.852i0.522 - 0.852i
Analytic cond. 2.794762.79476
Root an. cond. 1.671751.67175
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  + (0.0523 + 0.998i)2-s + (−1.45 − 1.18i)3-s + (−0.994 + 0.104i)4-s + (−1.98 − 1.03i)5-s + (1.10 − 1.51i)6-s + (0.182 + 2.63i)7-s + (−0.156 − 0.987i)8-s + (0.109 + 0.515i)9-s + (0.932 − 2.03i)10-s + (5.35 + 1.13i)11-s + (1.57 + 1.02i)12-s + (2.02 + 3.97i)13-s + (−2.62 + 0.320i)14-s + (1.66 + 3.85i)15-s + (0.978 − 0.207i)16-s + (1.59 + 0.612i)17-s + ⋯
L(s)  = 1  + (0.0370 + 0.706i)2-s + (−0.842 − 0.682i)3-s + (−0.497 + 0.0522i)4-s + (−0.885 − 0.464i)5-s + (0.450 − 0.620i)6-s + (0.0691 + 0.997i)7-s + (−0.0553 − 0.349i)8-s + (0.0365 + 0.171i)9-s + (0.294 − 0.642i)10-s + (1.61 + 0.342i)11-s + (0.454 + 0.295i)12-s + (0.561 + 1.10i)13-s + (−0.701 + 0.0857i)14-s + (0.429 + 0.995i)15-s + (0.244 − 0.0519i)16-s + (0.387 + 0.148i)17-s + ⋯

Functional equation

Λ(s)=(350s/2 ΓC(s) L(s)=((0.522−0.852i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.522 - 0.852i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(350s/2 ΓC(s+1/2) L(s)=((0.522−0.852i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.522 - 0.852i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 350350    =    2⋅52⋅72 \cdot 5^{2} \cdot 7
Sign: 0.522−0.852i0.522 - 0.852i
Analytic conductor: 2.794762.79476
Root analytic conductor: 1.671751.67175
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ350(283,⋅)\chi_{350} (283, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 350, ( :1/2), 0.522−0.852i)(2,\ 350,\ (\ :1/2),\ 0.522 - 0.852i)

Particular Values

L(1)L(1) ≈\approx 0.763610+0.427767i0.763610 + 0.427767i
L(12)L(\frac12) ≈\approx 0.763610+0.427767i0.763610 + 0.427767i
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)
bad2 1+(−0.0523−0.998i)T 1 + (-0.0523 - 0.998i)T
5 1+(1.98+1.03i)T 1 + (1.98 + 1.03i)T
7 1+(−0.182−2.63i)T 1 + (-0.182 - 2.63i)T
good3 1+(1.45+1.18i)T+(0.623+2.93i)T2 1 + (1.45 + 1.18i)T + (0.623 + 2.93i)T^{2}
11 1+(−5.35−1.13i)T+(10.0+4.47i)T2 1 + (-5.35 - 1.13i)T + (10.0 + 4.47i)T^{2}
13 1+(−2.02−3.97i)T+(−7.64+10.5i)T2 1 + (-2.02 - 3.97i)T + (-7.64 + 10.5i)T^{2}
17 1+(−1.59−0.612i)T+(12.6+11.3i)T2 1 + (-1.59 - 0.612i)T + (12.6 + 11.3i)T^{2}
19 1+(0.000953−0.00907i)T+(−18.5−3.95i)T2 1 + (0.000953 - 0.00907i)T + (-18.5 - 3.95i)T^{2}
23 1+(−4.72+0.247i)T+(22.8−2.40i)T2 1 + (-4.72 + 0.247i)T + (22.8 - 2.40i)T^{2}
29 1+(−1.47−2.03i)T+(−8.96+27.5i)T2 1 + (-1.47 - 2.03i)T + (-8.96 + 27.5i)T^{2}
31 1+(3.16+7.10i)T+(−20.7+23.0i)T2 1 + (3.16 + 7.10i)T + (-20.7 + 23.0i)T^{2}
37 1+(2.70−4.16i)T+(−15.0−33.8i)T2 1 + (2.70 - 4.16i)T + (-15.0 - 33.8i)T^{2}
41 1+(−4.63−1.50i)T+(33.1+24.0i)T2 1 + (-4.63 - 1.50i)T + (33.1 + 24.0i)T^{2}
43 1+(−5.48−5.48i)T+43iT2 1 + (-5.48 - 5.48i)T + 43iT^{2}
47 1+(2.82+7.36i)T+(−34.9+31.4i)T2 1 + (2.82 + 7.36i)T + (-34.9 + 31.4i)T^{2}
53 1+(6.66−8.22i)T+(−11.0−51.8i)T2 1 + (6.66 - 8.22i)T + (-11.0 - 51.8i)T^{2}
59 1+(−7.30−8.10i)T+(−6.16+58.6i)T2 1 + (-7.30 - 8.10i)T + (-6.16 + 58.6i)T^{2}
61 1+(2.29+2.07i)T+(6.37+60.6i)T2 1 + (2.29 + 2.07i)T + (6.37 + 60.6i)T^{2}
67 1+(−2.96+7.71i)T+(−49.7−44.8i)T2 1 + (-2.96 + 7.71i)T + (-49.7 - 44.8i)T^{2}
71 1+(−4.72+3.43i)T+(21.9−67.5i)T2 1 + (-4.72 + 3.43i)T + (21.9 - 67.5i)T^{2}
73 1+(9.66−6.27i)T+(29.6−66.6i)T2 1 + (9.66 - 6.27i)T + (29.6 - 66.6i)T^{2}
79 1+(−2.42+5.45i)T+(−52.8−58.7i)T2 1 + (-2.42 + 5.45i)T + (-52.8 - 58.7i)T^{2}
83 1+(−14.8+2.35i)T+(78.9−25.6i)T2 1 + (-14.8 + 2.35i)T + (78.9 - 25.6i)T^{2}
89 1+(−1.83+2.04i)T+(−9.30−88.5i)T2 1 + (-1.83 + 2.04i)T + (-9.30 - 88.5i)T^{2}
97 1+(−14.8−2.35i)T+(92.2+29.9i)T2 1 + (-14.8 - 2.35i)T + (92.2 + 29.9i)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

−11.79633211027495027810497190276, −11.21874854817299877651265539385, −9.233577646454955513234264656118, −8.946648756916195756389010150523, −7.66606188612989207419128675593, −6.68224215759424126759712739488, −6.07940927484201000317438197673, −4.86833973621051958577907764136, −3.75898690709365413567269764765, −1.29648886376238255454542446239, 0.837788760791178409790873756823, 3.38177677453793902581242168682, 4.03200988197061302915706635639, 5.13643849199985467110817193169, 6.43536153866113483689991312712, 7.55781134059754776040692751106, 8.694414537139781544315059759424, 9.876543619350294509301796618505, 10.88330928333222261121184970855, 10.99416096824099166065729933944

Graph of the ZZ-function along the critical line