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Properties

Label 2-3120-5.4-c1-0-62
Degree 22
Conductor 31203120
Sign −0.590+0.807i-0.590 + 0.807i
Analytic cond. 24.913324.9133
Root an. cond. 4.991324.99132
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  − i·3-s + (1.32 − 1.80i)5-s − 9-s + 4.64·11-s + i·13-s + (−1.80 − 1.32i)15-s − 4.24i·17-s − 6.24·19-s − 2.24i·23-s + (−1.51 − 4.76i)25-s + i·27-s + 9.21·29-s − 9.28·31-s − 4.64i·33-s − 7.28i·37-s + ⋯
L(s)  = 1  − 0.577i·3-s + (0.590 − 0.807i)5-s − 0.333·9-s + 1.39·11-s + 0.277i·13-s + (−0.466 − 0.340i)15-s − 1.03i·17-s − 1.43·19-s − 0.469i·23-s + (−0.303 − 0.952i)25-s + 0.192i·27-s + 1.71·29-s − 1.66·31-s − 0.807i·33-s − 1.19i·37-s + ⋯

Functional equation

Λ(s)=(3120s/2 ΓC(s) L(s)=((−0.590+0.807i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 3120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.590 + 0.807i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(3120s/2 ΓC(s+1/2) L(s)=((−0.590+0.807i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.590 + 0.807i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 31203120    =    24⋅3⋅5⋅132^{4} \cdot 3 \cdot 5 \cdot 13
Sign: −0.590+0.807i-0.590 + 0.807i
Analytic conductor: 24.913324.9133
Root analytic conductor: 4.991324.99132
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ3120(1249,⋅)\chi_{3120} (1249, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3120, ( :1/2), −0.590+0.807i)(2,\ 3120,\ (\ :1/2),\ -0.590 + 0.807i)

Particular Values

L(1)L(1) ≈\approx 1.8577685041.857768504
L(12)L(\frac12) ≈\approx 1.8577685041.857768504
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
3 1+iT 1 + iT
5 1+(−1.32+1.80i)T 1 + (-1.32 + 1.80i)T
13 1−iT 1 - iT
good7 1−7T2 1 - 7T^{2}
11 1−4.64T+11T2 1 - 4.64T + 11T^{2}
17 1+4.24iT−17T2 1 + 4.24iT - 17T^{2}
19 1+6.24T+19T2 1 + 6.24T + 19T^{2}
23 1+2.24iT−23T2 1 + 2.24iT - 23T^{2}
29 1−9.21T+29T2 1 - 9.21T + 29T^{2}
31 1+9.28T+31T2 1 + 9.28T + 31T^{2}
37 1+7.28iT−37T2 1 + 7.28iT - 37T^{2}
41 1+5.67T+41T2 1 + 5.67T + 41T^{2}
43 1−4.24iT−43T2 1 - 4.24iT - 43T^{2}
47 1−2.88iT−47T2 1 - 2.88iT - 47T^{2}
53 1+9.21iT−53T2 1 + 9.21iT - 53T^{2}
59 1−5.92T+59T2 1 - 5.92T + 59T^{2}
61 1−0.969T+61T2 1 - 0.969T + 61T^{2}
67 1+1.93iT−67T2 1 + 1.93iT - 67T^{2}
71 1+5.60T+71T2 1 + 5.60T + 71T^{2}
73 1+12.5iT−73T2 1 + 12.5iT - 73T^{2}
79 1−12.2T+79T2 1 - 12.2T + 79T^{2}
83 1−3.67iT−83T2 1 - 3.67iT - 83T^{2}
89 1−9.67T+89T2 1 - 9.67T + 89T^{2}
97 1−6iT−97T2 1 - 6iT - 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

−8.702160125020813854334287325133, −7.69070463292639961085106219391, −6.66900087466562694618685242862, −6.41164177992001988049353779881, −5.38808955063404529118705020526, −4.58876805687438348338886408060, −3.77789327831133070368561813568, −2.43635022999084056155742735816, −1.65211422586707982345360402373, −0.57406262764946145722890282107, 1.45748957006103370277085726975, 2.45703590356028158849151447077, 3.56479881321858100176329149335, 4.07280584953882041810949398383, 5.16021904409751230792705615748, 6.10597419909817265755560644654, 6.51155025867433857082019268527, 7.31305966201475698158461744232, 8.511780823662353010529048227511, 8.866125595092648902266858081011

Graph of the ZZ-function along the critical line