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Properties

Label 2-464-116.15-c1-0-1
Degree 22
Conductor 464464
Sign 0.00948−0.999i0.00948 - 0.999i
Analytic cond. 3.705053.70505
Root an. cond. 1.924851.92485
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  + (−1.77 − 2.82i)3-s + (0.925 + 0.738i)5-s + (−2.85 − 0.652i)7-s + (−3.52 + 7.31i)9-s + (−1.40 + 4.02i)11-s + (−0.443 − 0.920i)13-s + (0.442 − 3.92i)15-s + (−2.46 − 2.46i)17-s + (4.11 + 2.58i)19-s + (3.23 + 9.23i)21-s + (−4.63 + 3.69i)23-s + (−0.800 − 3.50i)25-s + (16.9 − 1.91i)27-s + (5.09 + 1.73i)29-s + (0.225 + 2.00i)31-s + ⋯
L(s)  = 1  + (−1.02 − 1.63i)3-s + (0.413 + 0.330i)5-s + (−1.08 − 0.246i)7-s + (−1.17 + 2.43i)9-s + (−0.424 + 1.21i)11-s + (−0.122 − 0.255i)13-s + (0.114 − 1.01i)15-s + (−0.597 − 0.597i)17-s + (0.944 + 0.593i)19-s + (0.704 + 2.01i)21-s + (−0.966 + 0.770i)23-s + (−0.160 − 0.701i)25-s + (3.26 − 0.367i)27-s + (0.946 + 0.322i)29-s + (0.0405 + 0.359i)31-s + ⋯

Functional equation

Λ(s)=(464s/2 ΓC(s) L(s)=((0.00948−0.999i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 464 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.00948 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(464s/2 ΓC(s+1/2) L(s)=((0.00948−0.999i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 464 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.00948 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 464464    =    24⋅292^{4} \cdot 29
Sign: 0.00948−0.999i0.00948 - 0.999i
Analytic conductor: 3.705053.70505
Root analytic conductor: 1.924851.92485
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ464(15,⋅)\chi_{464} (15, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 464, ( :1/2), 0.00948−0.999i)(2,\ 464,\ (\ :1/2),\ 0.00948 - 0.999i)

Particular Values

L(1)L(1) ≈\approx 0.160402+0.158889i0.160402 + 0.158889i
L(12)L(\frac12) ≈\approx 0.160402+0.158889i0.160402 + 0.158889i
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
29 1+(−5.09−1.73i)T 1 + (-5.09 - 1.73i)T
good3 1+(1.77+2.82i)T+(−1.30+2.70i)T2 1 + (1.77 + 2.82i)T + (-1.30 + 2.70i)T^{2}
5 1+(−0.925−0.738i)T+(1.11+4.87i)T2 1 + (-0.925 - 0.738i)T + (1.11 + 4.87i)T^{2}
7 1+(2.85+0.652i)T+(6.30+3.03i)T2 1 + (2.85 + 0.652i)T + (6.30 + 3.03i)T^{2}
11 1+(1.40−4.02i)T+(−8.60−6.85i)T2 1 + (1.40 - 4.02i)T + (-8.60 - 6.85i)T^{2}
13 1+(0.443+0.920i)T+(−8.10+10.1i)T2 1 + (0.443 + 0.920i)T + (-8.10 + 10.1i)T^{2}
17 1+(2.46+2.46i)T+17iT2 1 + (2.46 + 2.46i)T + 17iT^{2}
19 1+(−4.11−2.58i)T+(8.24+17.1i)T2 1 + (-4.11 - 2.58i)T + (8.24 + 17.1i)T^{2}
23 1+(4.63−3.69i)T+(5.11−22.4i)T2 1 + (4.63 - 3.69i)T + (5.11 - 22.4i)T^{2}
31 1+(−0.225−2.00i)T+(−30.2+6.89i)T2 1 + (-0.225 - 2.00i)T + (-30.2 + 6.89i)T^{2}
37 1+(10.8−3.78i)T+(28.9−23.0i)T2 1 + (10.8 - 3.78i)T + (28.9 - 23.0i)T^{2}
41 1+(4.85−4.85i)T−41iT2 1 + (4.85 - 4.85i)T - 41iT^{2}
43 1+(−4.38−0.493i)T+(41.9+9.56i)T2 1 + (-4.38 - 0.493i)T + (41.9 + 9.56i)T^{2}
47 1+(2.59+0.909i)T+(36.7+29.3i)T2 1 + (2.59 + 0.909i)T + (36.7 + 29.3i)T^{2}
53 1+(4.79−6.01i)T+(−11.7−51.6i)T2 1 + (4.79 - 6.01i)T + (-11.7 - 51.6i)T^{2}
59 1+4.30iT−59T2 1 + 4.30iT - 59T^{2}
61 1+(−3.63+2.28i)T+(26.4−54.9i)T2 1 + (-3.63 + 2.28i)T + (26.4 - 54.9i)T^{2}
67 1+(3.97+1.91i)T+(41.7+52.3i)T2 1 + (3.97 + 1.91i)T + (41.7 + 52.3i)T^{2}
71 1+(1.63−0.789i)T+(44.2−55.5i)T2 1 + (1.63 - 0.789i)T + (44.2 - 55.5i)T^{2}
73 1+(−2.98−0.336i)T+(71.1+16.2i)T2 1 + (-2.98 - 0.336i)T + (71.1 + 16.2i)T^{2}
79 1+(14.0−4.90i)T+(61.7−49.2i)T2 1 + (14.0 - 4.90i)T + (61.7 - 49.2i)T^{2}
83 1+(8.15−1.86i)T+(74.7−36.0i)T2 1 + (8.15 - 1.86i)T + (74.7 - 36.0i)T^{2}
89 1+(−6.41+0.722i)T+(86.7−19.8i)T2 1 + (-6.41 + 0.722i)T + (86.7 - 19.8i)T^{2}
97 1+(14.8+9.32i)T+(42.0+87.3i)T2 1 + (14.8 + 9.32i)T + (42.0 + 87.3i)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

−11.52608897603778859553812905209, −10.32806835848801746519251270281, −9.830292697964222507185518533832, −8.200739827288785709256911648931, −7.22971313021887820539222908655, −6.75100959489134594117708154344, −5.90317730332710378803588461888, −4.92047847768419609690097453783, −2.88488461190306930255125734212, −1.66440153165561715521856865331, 0.15159288531644226800590404508, 3.06218943604457533425007952151, 4.02886671743848293865800646509, 5.21637209434496936374605903719, 5.86305635908859472680843473838, 6.64732728554043345303154366896, 8.599195265210919257461555077637, 9.235165406604146269743621666522, 10.04450006110795694255321759352, 10.64757986501833265215258328937

Graph of the ZZ-function along the critical line