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Properties

Label 2-416-52.31-c1-0-12
Degree 22
Conductor 416416
Sign 0.105+0.994i0.105 + 0.994i
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  − 1.46i·3-s + (1.87 − 1.87i)5-s + (−0.675 + 0.675i)7-s + 0.865·9-s + (0.202 − 0.202i)11-s + (0.675 − 3.54i)13-s + (−2.74 − 2.74i)15-s + 0.539i·17-s + (1.66 + 1.66i)19-s + (0.987 + 0.987i)21-s − 4.67·23-s − 2.05i·25-s − 5.64i·27-s − 4.27·29-s + (−5.66 − 5.66i)31-s + ⋯
L(s)  = 1  − 0.843i·3-s + (0.840 − 0.840i)5-s + (−0.255 + 0.255i)7-s + 0.288·9-s + (0.0610 − 0.0610i)11-s + (0.187 − 0.982i)13-s + (−0.708 − 0.708i)15-s + 0.130i·17-s + (0.381 + 0.381i)19-s + (0.215 + 0.215i)21-s − 0.975·23-s − 0.411i·25-s − 1.08i·27-s − 0.793·29-s + (−1.01 − 1.01i)31-s + ⋯

Functional equation

Λ(s)=(416s/2 ΓC(s) L(s)=((0.105+0.994i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.105 + 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(416s/2 ΓC(s+1/2) L(s)=((0.105+0.994i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.105 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 416416    =    25⋅132^{5} \cdot 13
Sign: 0.105+0.994i0.105 + 0.994i
Analytic conductor: 3.321773.32177
Root analytic conductor: 1.822571.82257
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ416(31,⋅)\chi_{416} (31, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 416, ( :1/2), 0.105+0.994i)(2,\ 416,\ (\ :1/2),\ 0.105 + 0.994i)

Particular Values

L(1)L(1) ≈\approx 1.16345−1.04683i1.16345 - 1.04683i
L(12)L(\frac12) ≈\approx 1.16345−1.04683i1.16345 - 1.04683i
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+(−0.675+3.54i)T 1 + (-0.675 + 3.54i)T
good3 1+1.46iT−3T2 1 + 1.46iT - 3T^{2}
5 1+(−1.87+1.87i)T−5iT2 1 + (-1.87 + 1.87i)T - 5iT^{2}
7 1+(0.675−0.675i)T−7iT2 1 + (0.675 - 0.675i)T - 7iT^{2}
11 1+(−0.202+0.202i)T−11iT2 1 + (-0.202 + 0.202i)T - 11iT^{2}
17 1−0.539iT−17T2 1 - 0.539iT - 17T^{2}
19 1+(−1.66−1.66i)T+19iT2 1 + (-1.66 - 1.66i)T + 19iT^{2}
23 1+4.67T+23T2 1 + 4.67T + 23T^{2}
29 1+4.27T+29T2 1 + 4.27T + 29T^{2}
31 1+(5.66+5.66i)T+31iT2 1 + (5.66 + 5.66i)T + 31iT^{2}
37 1+(−4.80−4.80i)T+37iT2 1 + (-4.80 - 4.80i)T + 37iT^{2}
41 1+(−2.21+2.21i)T−41iT2 1 + (-2.21 + 2.21i)T - 41iT^{2}
43 1−12.5T+43T2 1 - 12.5T + 43T^{2}
47 1+(−1.08+1.08i)T−47iT2 1 + (-1.08 + 1.08i)T - 47iT^{2}
53 1−3.75T+53T2 1 - 3.75T + 53T^{2}
59 1+(3.41−3.41i)T−59iT2 1 + (3.41 - 3.41i)T - 59iT^{2}
61 1+12.6T+61T2 1 + 12.6T + 61T^{2}
67 1+(−9.28−9.28i)T+67iT2 1 + (-9.28 - 9.28i)T + 67iT^{2}
71 1+(−9.35−9.35i)T+71iT2 1 + (-9.35 - 9.35i)T + 71iT^{2}
73 1+(0.865+0.865i)T+73iT2 1 + (0.865 + 0.865i)T + 73iT^{2}
79 1−13.0iT−79T2 1 - 13.0iT - 79T^{2}
83 1+(3.12+3.12i)T+83iT2 1 + (3.12 + 3.12i)T + 83iT^{2}
89 1+(−6.21−6.21i)T+89iT2 1 + (-6.21 - 6.21i)T + 89iT^{2}
97 1+(7.67−7.67i)T−97iT2 1 + (7.67 - 7.67i)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

−11.03224177334233226873087137298, −9.893675527494367502486952634909, −9.277122760457355233903054772938, −8.130256295032696526488744842991, −7.39003014870538152324338170923, −6.03464451819148946773445799309, −5.57368512298982810965494114922, −4.06042218282149418101858970566, −2.36147677493962928224413752350, −1.12830976700908972340298573837, 2.02763062295960104066197859786, 3.47843798605697108462341812610, 4.46303085378377621533296059061, 5.73492316597641801199723614566, 6.68801935898461916320702187490, 7.55190177506152196743057163873, 9.239643231876698838438507005825, 9.517325926831623922685703245907, 10.56900839178756947667521755153, 11.00998775233045967003268331150

Graph of the ZZ-function along the critical line