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Properties

Label 2-352-88.5-c1-0-5
Degree 22
Conductor 352352
Sign 0.138+0.990i0.138 + 0.990i
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  + (−1.67 + 2.31i)3-s + (−2.48 − 0.805i)5-s + (0.158 − 0.115i)7-s + (−1.59 − 4.90i)9-s + (−2.89 − 1.62i)11-s + (3.78 − 1.22i)13-s + (6.02 − 4.37i)15-s + (−0.382 + 1.17i)17-s + (3.21 − 4.42i)19-s + 0.560i·21-s − 3.19·23-s + (1.45 + 1.05i)25-s + (5.87 + 1.90i)27-s + (1.78 + 2.45i)29-s + (−3.03 − 9.35i)31-s + ⋯
L(s)  = 1  + (−0.969 + 1.33i)3-s + (−1.10 − 0.360i)5-s + (0.0600 − 0.0436i)7-s + (−0.531 − 1.63i)9-s + (−0.871 − 0.490i)11-s + (1.04 − 0.340i)13-s + (1.55 − 1.13i)15-s + (−0.0928 + 0.285i)17-s + (0.738 − 1.01i)19-s + 0.122i·21-s − 0.666·23-s + (0.291 + 0.211i)25-s + (1.13 + 0.367i)27-s + (0.331 + 0.455i)29-s + (−0.545 − 1.67i)31-s + ⋯

Functional equation

Λ(s)=(352s/2 ΓC(s) L(s)=((0.138+0.990i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.138 + 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(352s/2 ΓC(s+1/2) L(s)=((0.138+0.990i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.138 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 352352    =    25⋅112^{5} \cdot 11
Sign: 0.138+0.990i0.138 + 0.990i
Analytic conductor: 2.810732.81073
Root analytic conductor: 1.676521.67652
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ352(49,⋅)\chi_{352} (49, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 352, ( :1/2), 0.138+0.990i)(2,\ 352,\ (\ :1/2),\ 0.138 + 0.990i)

Particular Values

L(1)L(1) ≈\approx 0.253211−0.220371i0.253211 - 0.220371i
L(12)L(\frac12) ≈\approx 0.253211−0.220371i0.253211 - 0.220371i
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+(2.89+1.62i)T 1 + (2.89 + 1.62i)T
good3 1+(1.67−2.31i)T+(−0.927−2.85i)T2 1 + (1.67 - 2.31i)T + (-0.927 - 2.85i)T^{2}
5 1+(2.48+0.805i)T+(4.04+2.93i)T2 1 + (2.48 + 0.805i)T + (4.04 + 2.93i)T^{2}
7 1+(−0.158+0.115i)T+(2.16−6.65i)T2 1 + (-0.158 + 0.115i)T + (2.16 - 6.65i)T^{2}
13 1+(−3.78+1.22i)T+(10.5−7.64i)T2 1 + (-3.78 + 1.22i)T + (10.5 - 7.64i)T^{2}
17 1+(0.382−1.17i)T+(−13.7−9.99i)T2 1 + (0.382 - 1.17i)T + (-13.7 - 9.99i)T^{2}
19 1+(−3.21+4.42i)T+(−5.87−18.0i)T2 1 + (-3.21 + 4.42i)T + (-5.87 - 18.0i)T^{2}
23 1+3.19T+23T2 1 + 3.19T + 23T^{2}
29 1+(−1.78−2.45i)T+(−8.96+27.5i)T2 1 + (-1.78 - 2.45i)T + (-8.96 + 27.5i)T^{2}
31 1+(3.03+9.35i)T+(−25.0+18.2i)T2 1 + (3.03 + 9.35i)T + (-25.0 + 18.2i)T^{2}
37 1+(4.43+6.10i)T+(−11.4+35.1i)T2 1 + (4.43 + 6.10i)T + (-11.4 + 35.1i)T^{2}
41 1+(5.91+4.30i)T+(12.6+38.9i)T2 1 + (5.91 + 4.30i)T + (12.6 + 38.9i)T^{2}
43 1−3.82iT−43T2 1 - 3.82iT - 43T^{2}
47 1+(0.410+0.298i)T+(14.5+44.6i)T2 1 + (0.410 + 0.298i)T + (14.5 + 44.6i)T^{2}
53 1+(−0.0202+0.00659i)T+(42.8−31.1i)T2 1 + (-0.0202 + 0.00659i)T + (42.8 - 31.1i)T^{2}
59 1+(3.31+4.56i)T+(−18.2+56.1i)T2 1 + (3.31 + 4.56i)T + (-18.2 + 56.1i)T^{2}
61 1+(12.3+4.00i)T+(49.3+35.8i)T2 1 + (12.3 + 4.00i)T + (49.3 + 35.8i)T^{2}
67 1−2.16iT−67T2 1 - 2.16iT - 67T^{2}
71 1+(1.31−4.06i)T+(−57.4−41.7i)T2 1 + (1.31 - 4.06i)T + (-57.4 - 41.7i)T^{2}
73 1+(−2.07+1.50i)T+(22.5−69.4i)T2 1 + (-2.07 + 1.50i)T + (22.5 - 69.4i)T^{2}
79 1+(−1.57−4.85i)T+(−63.9+46.4i)T2 1 + (-1.57 - 4.85i)T + (-63.9 + 46.4i)T^{2}
83 1+(−2.18−0.710i)T+(67.1+48.7i)T2 1 + (-2.18 - 0.710i)T + (67.1 + 48.7i)T^{2}
89 1+6.22T+89T2 1 + 6.22T + 89T^{2}
97 1+(0.257+0.792i)T+(−78.4+57.0i)T2 1 + (0.257 + 0.792i)T + (-78.4 + 57.0i)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

−11.10426296141263489237568687025, −10.67491610405270073783292122256, −9.547586776383980182083906456892, −8.555671778114868968677945369140, −7.61764681345048024786736361220, −6.07660431964673890113496187062, −5.21629631326852674260199617259, −4.25158821524304183053626283187, −3.38564987644838603532733279728, −0.26718298629666462678012045968, 1.59470394435154743765689264071, 3.40566530257385580711219363772, 4.95227721350620216895642054682, 6.06279672572496198544392325880, 7.00503120662548153972290036566, 7.70421920871280066103733149450, 8.461793631548944650963175391424, 10.21097384720534865901851766635, 11.04366957846045593182646715208, 11.94886573337401307118172179128

Graph of the ZZ-function along the critical line