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Properties

Label 2-2900-5.4-c1-0-35
Degree 22
Conductor 29002900
Sign −0.447+0.894i-0.447 + 0.894i
Analytic cond. 23.156623.1566
Root an. cond. 4.812134.81213
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  + 2.67i·3-s − 4.67i·7-s − 4.14·9-s − 0.672·11-s + 1.14i·13-s + 3.52i·17-s − 5.52·19-s + 12.4·21-s + 3.81i·23-s − 3.05i·27-s + 29-s − 1.52·31-s − 1.79i·33-s + 7.16i·37-s − 3.05·39-s + ⋯
L(s)  = 1  + 1.54i·3-s − 1.76i·7-s − 1.38·9-s − 0.202·11-s + 0.317i·13-s + 0.855i·17-s − 1.26·19-s + 2.72·21-s + 0.795i·23-s − 0.588i·27-s + 0.185·29-s − 0.274·31-s − 0.313i·33-s + 1.17i·37-s − 0.489·39-s + ⋯

Functional equation

Λ(s)=(2900s/2 ΓC(s) L(s)=((−0.447+0.894i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 2900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(2900s/2 ΓC(s+1/2) L(s)=((−0.447+0.894i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 29002900    =    22⋅52⋅292^{2} \cdot 5^{2} \cdot 29
Sign: −0.447+0.894i-0.447 + 0.894i
Analytic conductor: 23.156623.1566
Root analytic conductor: 4.812134.81213
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ2900(349,⋅)\chi_{2900} (349, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2900, ( :1/2), −0.447+0.894i)(2,\ 2900,\ (\ :1/2),\ -0.447 + 0.894i)

Particular Values

L(1)L(1) ≈\approx 0.10982660480.1098266048
L(12)L(\frac12) ≈\approx 0.10982660480.1098266048
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
29 1−T 1 - T
good3 1−2.67iT−3T2 1 - 2.67iT - 3T^{2}
7 1+4.67iT−7T2 1 + 4.67iT - 7T^{2}
11 1+0.672T+11T2 1 + 0.672T + 11T^{2}
13 1−1.14iT−13T2 1 - 1.14iT - 13T^{2}
17 1−3.52iT−17T2 1 - 3.52iT - 17T^{2}
19 1+5.52T+19T2 1 + 5.52T + 19T^{2}
23 1−3.81iT−23T2 1 - 3.81iT - 23T^{2}
31 1+1.52T+31T2 1 + 1.52T + 31T^{2}
37 1−7.16iT−37T2 1 - 7.16iT - 37T^{2}
41 1−2.85T+41T2 1 - 2.85T + 41T^{2}
43 1+8.96iT−43T2 1 + 8.96iT - 43T^{2}
47 1+6.67iT−47T2 1 + 6.67iT - 47T^{2}
53 1+10.4iT−53T2 1 + 10.4iT - 53T^{2}
59 1+10.7T+59T2 1 + 10.7T + 59T^{2}
61 1+14.4T+61T2 1 + 14.4T + 61T^{2}
67 1+7.81iT−67T2 1 + 7.81iT - 67T^{2}
71 1−4.48T+71T2 1 - 4.48T + 71T^{2}
73 1−4.96iT−73T2 1 - 4.96iT - 73T^{2}
79 1+2.38T+79T2 1 + 2.38T + 79T^{2}
83 1+14.0iT−83T2 1 + 14.0iT - 83T^{2}
89 1−1.63T+89T2 1 - 1.63T + 89T^{2}
97 1+9.32iT−97T2 1 + 9.32iT - 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

−8.616618635124189406701511058708, −7.88701510874141454017817595045, −6.99181270364461675739222540088, −6.21412767776938773074534029443, −5.13933422717018452734159081367, −4.40334797229183089283781543452, −3.90793296450129930832815466672, −3.25252600819871416931193780252, −1.68083076106882469784198718801, −0.03333993842817919259779111983, 1.43645747214057803097770483776, 2.53817237748939651516678700156, 2.73411741217272101909417238836, 4.45647266755760237515823561469, 5.45787165003908326172696923250, 6.13309846109282705300952856754, 6.59019743502478560804067157147, 7.65981540849304234871004620718, 8.075713882577139782989426050700, 8.953113208968614624873857441576

Graph of the ZZ-function along the critical line