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Properties

Label 2-2300-5.4-c1-0-28
Degree 22
Conductor 23002300
Sign −0.894+0.447i-0.894 + 0.447i
Analytic cond. 18.365518.3655
Root an. cond. 4.285504.28550
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  − 1.56i·3-s − 2.56i·7-s + 0.561·9-s + 2·11-s − 3.56i·13-s − 2.56i·17-s − 6·19-s − 4·21-s + i·23-s − 5.56i·27-s − 6.12·29-s + 7.24·31-s − 3.12i·33-s + 4.56i·37-s − 5.56·39-s + ⋯
L(s)  = 1  − 0.901i·3-s − 0.968i·7-s + 0.187·9-s + 0.603·11-s − 0.987i·13-s − 0.621i·17-s − 1.37·19-s − 0.872·21-s + 0.208i·23-s − 1.07i·27-s − 1.13·29-s + 1.30·31-s − 0.543i·33-s + 0.749i·37-s − 0.890·39-s + ⋯

Functional equation

Λ(s)=(2300s/2 ΓC(s) L(s)=((−0.894+0.447i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 2300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(2300s/2 ΓC(s+1/2) L(s)=((−0.894+0.447i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 23002300    =    22⋅52⋅232^{2} \cdot 5^{2} \cdot 23
Sign: −0.894+0.447i-0.894 + 0.447i
Analytic conductor: 18.365518.3655
Root analytic conductor: 4.285504.28550
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ2300(1749,⋅)\chi_{2300} (1749, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2300, ( :1/2), −0.894+0.447i)(2,\ 2300,\ (\ :1/2),\ -0.894 + 0.447i)

Particular Values

L(1)L(1) ≈\approx 1.5192733751.519273375
L(12)L(\frac12) ≈\approx 1.5192733751.519273375
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 1
23 1−iT 1 - iT
good3 1+1.56iT−3T2 1 + 1.56iT - 3T^{2}
7 1+2.56iT−7T2 1 + 2.56iT - 7T^{2}
11 1−2T+11T2 1 - 2T + 11T^{2}
13 1+3.56iT−13T2 1 + 3.56iT - 13T^{2}
17 1+2.56iT−17T2 1 + 2.56iT - 17T^{2}
19 1+6T+19T2 1 + 6T + 19T^{2}
29 1+6.12T+29T2 1 + 6.12T + 29T^{2}
31 1−7.24T+31T2 1 - 7.24T + 31T^{2}
37 1−4.56iT−37T2 1 - 4.56iT - 37T^{2}
41 1−4.12T+41T2 1 - 4.12T + 41T^{2}
43 1−43T2 1 - 43T^{2}
47 1+4.68iT−47T2 1 + 4.68iT - 47T^{2}
53 1+4.56iT−53T2 1 + 4.56iT - 53T^{2}
59 1−3.68T+59T2 1 - 3.68T + 59T^{2}
61 1+7.12T+61T2 1 + 7.12T + 61T^{2}
67 1−8.56iT−67T2 1 - 8.56iT - 67T^{2}
71 1−10.1T+71T2 1 - 10.1T + 71T^{2}
73 1+4.43iT−73T2 1 + 4.43iT - 73T^{2}
79 1+4.87T+79T2 1 + 4.87T + 79T^{2}
83 1+13.9iT−83T2 1 + 13.9iT - 83T^{2}
89 1+14.2T+89T2 1 + 14.2T + 89T^{2}
97 1−13.1iT−97T2 1 - 13.1iT - 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

−8.425794017452273794151957018317, −7.84735525707203661722662645031, −7.06745213933410903875481448899, −6.59907380602805694765537131085, −5.69174898879537983679204803186, −4.54208524705766411609994900361, −3.83383714250876091929595743266, −2.65485863579735171951860026419, −1.51112087306479009439007422655, −0.52818152986298899885087905296, 1.64604722216838827234324406590, 2.62845434297040578089079416770, 4.00428535033584133867341723346, 4.26527282186323017147359551419, 5.34506170508296469213931086513, 6.20157648963508884697311454059, 6.83837185917807805546585086895, 7.983822562655058200323374638573, 8.843496249166157591224255948331, 9.259432319732943686558634576995

Graph of the ZZ-function along the critical line