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Properties

Label 2-2320-29.28-c1-0-26
Degree 22
Conductor 23202320
Sign 0.964+0.262i0.964 + 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 + 2.73·7-s + 0.999·9-s − 0.378i·11-s + 5.46·13-s + 1.41i·15-s + 3.48i·17-s + 4.24i·19-s − 3.86i·21-s − 2.19·23-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 + 1.03·7-s + 0.333·9-s − 0.114i·11-s + 1.51·13-s + 0.365i·15-s + 0.845i·17-s + 0.973i·19-s − 0.843i·21-s − 0.457·23-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+0.262i0.964 + 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 2.1404658802.140465880
L(12)L(\frac12) ≈\approx 2.1404658802.140465880
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 1−2.73T+7T2 1 - 2.73T + 7T^{2}
11 1+0.378iT−11T2 1 + 0.378iT - 11T^{2}
13 1−5.46T+13T2 1 - 5.46T + 13T^{2}
17 1−3.48iT−17T2 1 - 3.48iT - 17T^{2}
19 1−4.24iT−19T2 1 - 4.24iT - 19T^{2}
23 1+2.19T+23T2 1 + 2.19T + 23T^{2}
31 1−4.24iT−31T2 1 - 4.24iT - 31T^{2}
37 1−4.24iT−37T2 1 - 4.24iT - 37T^{2}
41 1−5.93iT−41T2 1 - 5.93iT - 41T^{2}
43 1−4.24iT−43T2 1 - 4.24iT - 43T^{2}
47 1−11.2iT−47T2 1 - 11.2iT - 47T^{2}
53 1+53T2 1 + 53T^{2}
59 1+6T+59T2 1 + 6T + 59T^{2}
61 1+11.5iT−61T2 1 + 11.5iT - 61T^{2}
67 1−13.1T+67T2 1 - 13.1T + 67T^{2}
71 1+6T+71T2 1 + 6T + 71T^{2}
73 1+15.8iT−73T2 1 + 15.8iT - 73T^{2}
79 1+1.13iT−79T2 1 + 1.13iT - 79T^{2}
83 1−8.19T+83T2 1 - 8.19T + 83T^{2}
89 1−7.72iT−89T2 1 - 7.72iT - 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

−8.565831099601683828653970150659, −8.087375977248019769396706986620, −7.71473948602675444592189910973, −6.46360284952031574797069357621, −6.18529818981322819974045782285, −4.90955088820831404944897835855, −4.13162452186157690293185474822, −3.21001717308991377220075045661, −1.69938768256210903696223887381, −1.21745934827391173854219011712, 0.907199724828692192729413359244, 2.23263135057711408546083913699, 3.54712748094132065713288754443, 4.21395927081618065266961046696, 4.89365679511686793205808391214, 5.69633403724829802711995551895, 6.83907882159094368529177608496, 7.51108649733405599930184536904, 8.479970438063116086206922405722, 8.875993741515632467892366075239

Graph of the ZZ-function along the critical line