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Properties

Label 2-352-88.19-c1-0-6
Degree 22
Conductor 352352
Sign 0.966−0.254i0.966 - 0.254i
Analytic cond. 2.810732.81073
Root an. cond. 1.676521.67652
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  + (0.625 + 1.92i)3-s + (1.75 − 2.41i)5-s + (0.216 − 0.667i)7-s + (−0.888 + 0.645i)9-s + (2.75 − 1.85i)11-s + (2.50 − 1.82i)13-s + (5.73 + 1.86i)15-s + (−2.54 + 3.49i)17-s + (−4.99 + 1.62i)19-s + 1.42·21-s + 0.883i·23-s + (−1.19 − 3.68i)25-s + (3.11 + 2.26i)27-s + (−3.13 + 9.66i)29-s + (−4.71 − 6.48i)31-s + ⋯
L(s)  = 1  + (0.361 + 1.11i)3-s + (0.783 − 1.07i)5-s + (0.0819 − 0.252i)7-s + (−0.296 + 0.215i)9-s + (0.829 − 0.558i)11-s + (0.695 − 0.505i)13-s + (1.48 + 0.481i)15-s + (−0.616 + 0.848i)17-s + (−1.14 + 0.372i)19-s + 0.310·21-s + 0.184i·23-s + (−0.239 − 0.737i)25-s + (0.599 + 0.435i)27-s + (−0.583 + 1.79i)29-s + (−0.846 − 1.16i)31-s + ⋯

Functional equation

Λ(s)=(352s/2 ΓC(s) L(s)=((0.966−0.254i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.966 - 0.254i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(352s/2 ΓC(s+1/2) L(s)=((0.966−0.254i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.966 - 0.254i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 352352    =    25⋅112^{5} \cdot 11
Sign: 0.966−0.254i0.966 - 0.254i
Analytic conductor: 2.810732.81073
Root analytic conductor: 1.676521.67652
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ352(239,⋅)\chi_{352} (239, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 352, ( :1/2), 0.966−0.254i)(2,\ 352,\ (\ :1/2),\ 0.966 - 0.254i)

Particular Values

L(1)L(1) ≈\approx 1.73755+0.225137i1.73755 + 0.225137i
L(12)L(\frac12) ≈\approx 1.73755+0.225137i1.73755 + 0.225137i
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 1
11 1+(−2.75+1.85i)T 1 + (-2.75 + 1.85i)T
good3 1+(−0.625−1.92i)T+(−2.42+1.76i)T2 1 + (-0.625 - 1.92i)T + (-2.42 + 1.76i)T^{2}
5 1+(−1.75+2.41i)T+(−1.54−4.75i)T2 1 + (-1.75 + 2.41i)T + (-1.54 - 4.75i)T^{2}
7 1+(−0.216+0.667i)T+(−5.66−4.11i)T2 1 + (-0.216 + 0.667i)T + (-5.66 - 4.11i)T^{2}
13 1+(−2.50+1.82i)T+(4.01−12.3i)T2 1 + (-2.50 + 1.82i)T + (4.01 - 12.3i)T^{2}
17 1+(2.54−3.49i)T+(−5.25−16.1i)T2 1 + (2.54 - 3.49i)T + (-5.25 - 16.1i)T^{2}
19 1+(4.99−1.62i)T+(15.3−11.1i)T2 1 + (4.99 - 1.62i)T + (15.3 - 11.1i)T^{2}
23 1−0.883iT−23T2 1 - 0.883iT - 23T^{2}
29 1+(3.13−9.66i)T+(−23.4−17.0i)T2 1 + (3.13 - 9.66i)T + (-23.4 - 17.0i)T^{2}
31 1+(4.71+6.48i)T+(−9.57+29.4i)T2 1 + (4.71 + 6.48i)T + (-9.57 + 29.4i)T^{2}
37 1+(−2.84−0.924i)T+(29.9+21.7i)T2 1 + (-2.84 - 0.924i)T + (29.9 + 21.7i)T^{2}
41 1+(−2.79+0.906i)T+(33.1−24.0i)T2 1 + (-2.79 + 0.906i)T + (33.1 - 24.0i)T^{2}
43 1−2.57iT−43T2 1 - 2.57iT - 43T^{2}
47 1+(4.57−1.48i)T+(38.0−27.6i)T2 1 + (4.57 - 1.48i)T + (38.0 - 27.6i)T^{2}
53 1+(7.22+9.94i)T+(−16.3+50.4i)T2 1 + (7.22 + 9.94i)T + (-16.3 + 50.4i)T^{2}
59 1+(−1.63+5.02i)T+(−47.7−34.6i)T2 1 + (-1.63 + 5.02i)T + (-47.7 - 34.6i)T^{2}
61 1+(6.21+4.51i)T+(18.8+58.0i)T2 1 + (6.21 + 4.51i)T + (18.8 + 58.0i)T^{2}
67 1−5.11T+67T2 1 - 5.11T + 67T^{2}
71 1+(4.66−6.42i)T+(−21.9−67.5i)T2 1 + (4.66 - 6.42i)T + (-21.9 - 67.5i)T^{2}
73 1+(4.05+1.31i)T+(59.0+42.9i)T2 1 + (4.05 + 1.31i)T + (59.0 + 42.9i)T^{2}
79 1+(−2.28+1.65i)T+(24.4−75.1i)T2 1 + (-2.28 + 1.65i)T + (24.4 - 75.1i)T^{2}
83 1+(−0.679+0.934i)T+(−25.6−78.9i)T2 1 + (-0.679 + 0.934i)T + (-25.6 - 78.9i)T^{2}
89 1+9.59T+89T2 1 + 9.59T + 89T^{2}
97 1+(−5.12+3.72i)T+(29.9−92.2i)T2 1 + (-5.12 + 3.72i)T + (29.9 - 92.2i)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

−11.18100502914202855208399116357, −10.54989310477775856610883859033, −9.465329189986587773793941456075, −8.961349743521170157624431041098, −8.203791195402211713557395344469, −6.45907948155804718256140043234, −5.50972315383922853435973378770, −4.37386998652082777272479551527, −3.57187090523784214348679985546, −1.55570748409037766801217136484, 1.79848368273680931550693282686, 2.61896264132192363270788793994, 4.29992759771187308770980752143, 6.06845546696229875963809803602, 6.69977627972717519369523907722, 7.37643915467477048647960522106, 8.660225747443238524536072589089, 9.488867630987447717537945479207, 10.61392512375839290457841242160, 11.46420773750072091852846420065

Graph of the ZZ-function along the critical line