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Properties

Label 2-416-104.99-c1-0-5
Degree 22
Conductor 416416
Sign 0.876−0.481i0.876 - 0.481i
Analytic cond. 3.321773.32177
Root an. cond. 1.822571.82257
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  + 2.95·3-s + (−1.04 + 1.04i)5-s + (2.37 + 2.37i)7-s + 5.71·9-s + (−2.03 + 2.03i)11-s + (−3.25 − 1.55i)13-s + (−3.08 + 3.08i)15-s − 4.25i·17-s + (−0.214 − 0.214i)19-s + (7.00 + 7.00i)21-s − 0.169·23-s + 2.82i·25-s + 8.01·27-s − 9.09i·29-s + (4.96 − 4.96i)31-s + ⋯
L(s)  = 1  + 1.70·3-s + (−0.466 + 0.466i)5-s + (0.896 + 0.896i)7-s + 1.90·9-s + (−0.614 + 0.614i)11-s + (−0.902 − 0.430i)13-s + (−0.795 + 0.795i)15-s − 1.03i·17-s + (−0.0492 − 0.0492i)19-s + (1.52 + 1.52i)21-s − 0.0353·23-s + 0.564i·25-s + 1.54·27-s − 1.68i·29-s + (0.892 − 0.892i)31-s + ⋯

Functional equation

Λ(s)=(416s/2 ΓC(s) L(s)=((0.876−0.481i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.876 - 0.481i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(416s/2 ΓC(s+1/2) L(s)=((0.876−0.481i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.876 - 0.481i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 416416    =    25⋅132^{5} \cdot 13
Sign: 0.876−0.481i0.876 - 0.481i
Analytic conductor: 3.321773.32177
Root analytic conductor: 1.822571.82257
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ416(47,⋅)\chi_{416} (47, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 416, ( :1/2), 0.876−0.481i)(2,\ 416,\ (\ :1/2),\ 0.876 - 0.481i)

Particular Values

L(1)L(1) ≈\approx 2.19573+0.563687i2.19573 + 0.563687i
L(12)L(\frac12) ≈\approx 2.19573+0.563687i2.19573 + 0.563687i
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
13 1+(3.25+1.55i)T 1 + (3.25 + 1.55i)T
good3 1−2.95T+3T2 1 - 2.95T + 3T^{2}
5 1+(1.04−1.04i)T−5iT2 1 + (1.04 - 1.04i)T - 5iT^{2}
7 1+(−2.37−2.37i)T+7iT2 1 + (-2.37 - 2.37i)T + 7iT^{2}
11 1+(2.03−2.03i)T−11iT2 1 + (2.03 - 2.03i)T - 11iT^{2}
17 1+4.25iT−17T2 1 + 4.25iT - 17T^{2}
19 1+(0.214+0.214i)T+19iT2 1 + (0.214 + 0.214i)T + 19iT^{2}
23 1+0.169T+23T2 1 + 0.169T + 23T^{2}
29 1+9.09iT−29T2 1 + 9.09iT - 29T^{2}
31 1+(−4.96+4.96i)T−31iT2 1 + (-4.96 + 4.96i)T - 31iT^{2}
37 1+(−2.37−2.37i)T+37iT2 1 + (-2.37 - 2.37i)T + 37iT^{2}
41 1+(3.95+3.95i)T+41iT2 1 + (3.95 + 3.95i)T + 41iT^{2}
43 1+2.48iT−43T2 1 + 2.48iT - 43T^{2}
47 1+(0.869+0.869i)T+47iT2 1 + (0.869 + 0.869i)T + 47iT^{2}
53 1+1.81iT−53T2 1 + 1.81iT - 53T^{2}
59 1+(3.97−3.97i)T−59iT2 1 + (3.97 - 3.97i)T - 59iT^{2}
61 1+0.851iT−61T2 1 + 0.851iT - 61T^{2}
67 1+(−8.69−8.69i)T+67iT2 1 + (-8.69 - 8.69i)T + 67iT^{2}
71 1+(3.22−3.22i)T−71iT2 1 + (3.22 - 3.22i)T - 71iT^{2}
73 1+(−3.95+3.95i)T−73iT2 1 + (-3.95 + 3.95i)T - 73iT^{2}
79 1−10.5iT−79T2 1 - 10.5iT - 79T^{2}
83 1+(2.18+2.18i)T+83iT2 1 + (2.18 + 2.18i)T + 83iT^{2}
89 1+(7.97−7.97i)T−89iT2 1 + (7.97 - 7.97i)T - 89iT^{2}
97 1+(10.2+10.2i)T+97iT2 1 + (10.2 + 10.2i)T + 97iT^{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

−11.39536636452434121602358442029, −10.03029521892214285013450903954, −9.452311427207470583199716238391, −8.330740673066742317533480236868, −7.82721491020735642140184376650, −7.08811388975195452514425115371, −5.28944971660766444802435797511, −4.22515270905449091683005460553, −2.80254389216636100000346603216, −2.24725068038829795829841998935, 1.56522268946033502653853743968, 2.98501701026836102600151570476, 4.09811097164885429687007385768, 4.90831316115699694386381336803, 6.83665695936558015072856253146, 7.910419510279208246362431087752, 8.185723273702117523298059753860, 9.071484187010618442147579144928, 10.16958385158856202415439556359, 10.92860907404020328981603924488

Graph of the ZZ-function along the critical line