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Properties

Label 2-2320-29.28-c1-0-9
Degree 22
Conductor 23202320
Sign −0.964−0.262i-0.964 - 0.262i
Analytic cond. 18.525218.5252
Root an. cond. 4.304104.30410
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  + 1.41i·3-s − 5-s − 0.732·7-s + 0.999·9-s + 5.27i·11-s − 1.46·13-s − 1.41i·15-s + 6.31i·17-s − 4.24i·19-s − 1.03i·21-s + 8.19·23-s + 25-s + 5.65i·27-s + (−5.19 − 1.41i)29-s − 4.24i·31-s + ⋯
L(s)  = 1  + 0.816i·3-s − 0.447·5-s − 0.276·7-s + 0.333·9-s + 1.59i·11-s − 0.406·13-s − 0.365i·15-s + 1.53i·17-s − 0.973i·19-s − 0.225i·21-s + 1.70·23-s + 0.200·25-s + 1.08i·27-s + (−0.964 − 0.262i)29-s − 0.762i·31-s + ⋯

Functional equation

Λ(s)=(2320s/2 ΓC(s) L(s)=((−0.964−0.262i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 2320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.964 - 0.262i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(2320s/2 ΓC(s+1/2) L(s)=((−0.964−0.262i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.964 - 0.262i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 23202320    =    24⋅5⋅292^{4} \cdot 5 \cdot 29
Sign: −0.964−0.262i-0.964 - 0.262i
Analytic conductor: 18.525218.5252
Root analytic conductor: 4.304104.30410
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ2320(1681,⋅)\chi_{2320} (1681, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2320, ( :1/2), −0.964−0.262i)(2,\ 2320,\ (\ :1/2),\ -0.964 - 0.262i)

Particular Values

L(1)L(1) ≈\approx 1.0740245771.074024577
L(12)L(\frac12) ≈\approx 1.0740245771.074024577
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+T 1 + T
29 1+(5.19+1.41i)T 1 + (5.19 + 1.41i)T
good3 1−1.41iT−3T2 1 - 1.41iT - 3T^{2}
7 1+0.732T+7T2 1 + 0.732T + 7T^{2}
11 1−5.27iT−11T2 1 - 5.27iT - 11T^{2}
13 1+1.46T+13T2 1 + 1.46T + 13T^{2}
17 1−6.31iT−17T2 1 - 6.31iT - 17T^{2}
19 1+4.24iT−19T2 1 + 4.24iT - 19T^{2}
23 1−8.19T+23T2 1 - 8.19T + 23T^{2}
31 1+4.24iT−31T2 1 + 4.24iT - 31T^{2}
37 1+4.24iT−37T2 1 + 4.24iT - 37T^{2}
41 1−8.76iT−41T2 1 - 8.76iT - 41T^{2}
43 1+4.24iT−43T2 1 + 4.24iT - 43T^{2}
47 1−8.38iT−47T2 1 - 8.38iT - 47T^{2}
53 1+53T2 1 + 53T^{2}
59 1+6T+59T2 1 + 6T + 59T^{2}
61 1+3.10iT−61T2 1 + 3.10iT - 61T^{2}
67 1+11.1T+67T2 1 + 11.1T + 67T^{2}
71 1+6T+71T2 1 + 6T + 71T^{2}
73 1−1.13iT−73T2 1 - 1.13iT - 73T^{2}
79 1−15.8iT−79T2 1 - 15.8iT - 79T^{2}
83 1+2.19T+83T2 1 + 2.19T + 83T^{2}
89 1−2.07iT−89T2 1 - 2.07iT - 89T^{2}
97 1−7.34iT−97T2 1 - 7.34iT - 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

−9.517524711980359789882062516417, −8.782473678187901703555732287381, −7.64573223523625302878571476942, −7.18078314200113158957783414528, −6.30836780544580400354980643314, −5.10249504617179500395950285419, −4.50268838607527986336042497483, −3.88093688066149253548001911777, −2.79258800791675223531111938517, −1.55886991044880893217265483730, 0.38112875905549391131519117578, 1.44546909418293370403047546921, 2.87627109242000962103174978965, 3.47735647062810129913086335722, 4.72728287401429930062566816235, 5.55728646963653702755526927768, 6.47326237654290066674501015379, 7.20795023212027237653289138414, 7.67529897030406957215542048350, 8.646326429080559545066155399714

Graph of the ZZ-function along the critical line