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Properties

Label 2-416-104.83-c1-0-6
Degree 22
Conductor 416416
Sign 0.957−0.289i0.957 - 0.289i
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  + 3-s + (2.54 + 2.54i)5-s + (2.54 − 2.54i)7-s − 2·9-s + (−1 − i)11-s + (2.54 + 2.54i)13-s + (2.54 + 2.54i)15-s − 3i·17-s + (−2 + 2i)19-s + (2.54 − 2.54i)21-s − 5.09·23-s + 7.99i·25-s − 5·27-s − 5.09i·29-s + (5.09 + 5.09i)31-s + ⋯
L(s)  = 1  + 0.577·3-s + (1.14 + 1.14i)5-s + (0.963 − 0.963i)7-s − 0.666·9-s + (−0.301 − 0.301i)11-s + (0.707 + 0.707i)13-s + (0.658 + 0.658i)15-s − 0.727i·17-s + (−0.458 + 0.458i)19-s + (0.556 − 0.556i)21-s − 1.06·23-s + 1.59i·25-s − 0.962·27-s − 0.946i·29-s + (0.915 + 0.915i)31-s + ⋯

Functional equation

Λ(s)=(416s/2 ΓC(s) L(s)=((0.957−0.289i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.957 - 0.289i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(416s/2 ΓC(s+1/2) L(s)=((0.957−0.289i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.957 - 0.289i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 416416    =    25⋅132^{5} \cdot 13
Sign: 0.957−0.289i0.957 - 0.289i
Analytic conductor: 3.321773.32177
Root analytic conductor: 1.822571.82257
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ416(239,⋅)\chi_{416} (239, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 416, ( :1/2), 0.957−0.289i)(2,\ 416,\ (\ :1/2),\ 0.957 - 0.289i)

Particular Values

L(1)L(1) ≈\approx 1.98980+0.294628i1.98980 + 0.294628i
L(12)L(\frac12) ≈\approx 1.98980+0.294628i1.98980 + 0.294628i
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+(−2.54−2.54i)T 1 + (-2.54 - 2.54i)T
good3 1−T+3T2 1 - T + 3T^{2}
5 1+(−2.54−2.54i)T+5iT2 1 + (-2.54 - 2.54i)T + 5iT^{2}
7 1+(−2.54+2.54i)T−7iT2 1 + (-2.54 + 2.54i)T - 7iT^{2}
11 1+(1+i)T+11iT2 1 + (1 + i)T + 11iT^{2}
17 1+3iT−17T2 1 + 3iT - 17T^{2}
19 1+(2−2i)T−19iT2 1 + (2 - 2i)T - 19iT^{2}
23 1+5.09T+23T2 1 + 5.09T + 23T^{2}
29 1+5.09iT−29T2 1 + 5.09iT - 29T^{2}
31 1+(−5.09−5.09i)T+31iT2 1 + (-5.09 - 5.09i)T + 31iT^{2}
37 1+(2.54−2.54i)T−37iT2 1 + (2.54 - 2.54i)T - 37iT^{2}
41 1+(−6+6i)T−41iT2 1 + (-6 + 6i)T - 41iT^{2}
43 1−iT−43T2 1 - iT - 43T^{2}
47 1+(−2.54+2.54i)T−47iT2 1 + (-2.54 + 2.54i)T - 47iT^{2}
53 1−5.09iT−53T2 1 - 5.09iT - 53T^{2}
59 1+(8+8i)T+59iT2 1 + (8 + 8i)T + 59iT^{2}
61 1−61T2 1 - 61T^{2}
67 1+(3−3i)T−67iT2 1 + (3 - 3i)T - 67iT^{2}
71 1+(7.64+7.64i)T+71iT2 1 + (7.64 + 7.64i)T + 71iT^{2}
73 1+(6+6i)T+73iT2 1 + (6 + 6i)T + 73iT^{2}
79 1−5.09iT−79T2 1 - 5.09iT - 79T^{2}
83 1+(5−5i)T−83iT2 1 + (5 - 5i)T - 83iT^{2}
89 1+(2+2i)T+89iT2 1 + (2 + 2i)T + 89iT^{2}
97 1+(7−7i)T−97iT2 1 + (7 - 7i)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

−10.98978815745216941247365647574, −10.47378333256809716956113449034, −9.522846882555152077004091591682, −8.460833272230017705349779398442, −7.60721889227404273118802203977, −6.53306096055565267878808075195, −5.68900522642042334314257742942, −4.21350629763050392852135877784, −2.93739711415506004510821247999, −1.83594166905495058642333256099, 1.64159201182266667141281234590, 2.63960168777552721901613107640, 4.44283662951956588683922956516, 5.56051991924676673029888604411, 5.98747868371912356386435455814, 7.949970144349250383813649320626, 8.546476828639421592298622594852, 9.075571284278865699110782216726, 10.11312287058664130854485052809, 11.17376927509350179159868371787

Graph of the ZZ-function along the critical line