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Properties

Label 2-1520-5.4-c1-0-16
Degree 22
Conductor 15201520
Sign 0.316−0.948i0.316 - 0.948i
Analytic cond. 12.137212.1372
Root an. cond. 3.483853.48385
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  + 0.414i·3-s + (0.707 − 2.12i)5-s + 4.41i·7-s + 2.82·9-s + 1.41·11-s + 5.82i·13-s + (0.878 + 0.292i)15-s − i·17-s − 19-s − 1.82·21-s − 0.757i·23-s + (−3.99 − 3i)25-s + 2.41i·27-s + 0.171·29-s − 6.24·31-s + ⋯
L(s)  = 1  + 0.239i·3-s + (0.316 − 0.948i)5-s + 1.66i·7-s + 0.942·9-s + 0.426·11-s + 1.61i·13-s + (0.226 + 0.0756i)15-s − 0.242i·17-s − 0.229·19-s − 0.398·21-s − 0.157i·23-s + (−0.799 − 0.600i)25-s + 0.464i·27-s + 0.0318·29-s − 1.12·31-s + ⋯

Functional equation

Λ(s)=(1520s/2 ΓC(s) L(s)=((0.316−0.948i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.316 - 0.948i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1520s/2 ΓC(s+1/2) L(s)=((0.316−0.948i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.316 - 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 15201520    =    24⋅5⋅192^{4} \cdot 5 \cdot 19
Sign: 0.316−0.948i0.316 - 0.948i
Analytic conductor: 12.137212.1372
Root analytic conductor: 3.483853.48385
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1520(609,⋅)\chi_{1520} (609, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1520, ( :1/2), 0.316−0.948i)(2,\ 1520,\ (\ :1/2),\ 0.316 - 0.948i)

Particular Values

L(1)L(1) ≈\approx 1.8101427451.810142745
L(12)L(\frac12) ≈\approx 1.8101427451.810142745
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
5 1+(−0.707+2.12i)T 1 + (-0.707 + 2.12i)T
19 1+T 1 + T
good3 1−0.414iT−3T2 1 - 0.414iT - 3T^{2}
7 1−4.41iT−7T2 1 - 4.41iT - 7T^{2}
11 1−1.41T+11T2 1 - 1.41T + 11T^{2}
13 1−5.82iT−13T2 1 - 5.82iT - 13T^{2}
17 1+iT−17T2 1 + iT - 17T^{2}
23 1+0.757iT−23T2 1 + 0.757iT - 23T^{2}
29 1−0.171T+29T2 1 - 0.171T + 29T^{2}
31 1+6.24T+31T2 1 + 6.24T + 31T^{2}
37 1−8.48iT−37T2 1 - 8.48iT - 37T^{2}
41 1+4.24T+41T2 1 + 4.24T + 41T^{2}
43 1+1.75iT−43T2 1 + 1.75iT - 43T^{2}
47 1−47T2 1 - 47T^{2}
53 1−5.48iT−53T2 1 - 5.48iT - 53T^{2}
59 1−6.89T+59T2 1 - 6.89T + 59T^{2}
61 1−14.2T+61T2 1 - 14.2T + 61T^{2}
67 1−4.75iT−67T2 1 - 4.75iT - 67T^{2}
71 1−13.4T+71T2 1 - 13.4T + 71T^{2}
73 1−11.4iT−73T2 1 - 11.4iT - 73T^{2}
79 1+6.48T+79T2 1 + 6.48T + 79T^{2}
83 1+14.4iT−83T2 1 + 14.4iT - 83T^{2}
89 1+7.07T+89T2 1 + 7.07T + 89T^{2}
97 1−0.343iT−97T2 1 - 0.343iT - 97T^{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

−9.483604460314060470062975422760, −8.888262113162242075935466997130, −8.414070790121688220854069250812, −7.08136136242458489342506515435, −6.32502142692066894376224366107, −5.38861425307692806763428886269, −4.69878047542539799213847256189, −3.83994115418674686283762026751, −2.29581768583533008892180805253, −1.53091810429916511590236429102, 0.74638202948797869613309201616, 2.00809282202571200259223964157, 3.46681132702447006265933140103, 3.92699870702487615962091440827, 5.19670382211094827126225664815, 6.24842408461878396296566214830, 7.09657347038480007429552401362, 7.41988343917465567438021961684, 8.275902157915052970183863972050, 9.674993082000573170215335481647

Graph of the ZZ-function along the critical line