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Properties

Label 2-3200-8.5-c1-0-22
Degree 22
Conductor 32003200
Sign −i-i
Analytic cond. 25.552125.5521
Root an. cond. 5.054915.05491
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 − 3.16·7-s + 0.999·9-s − 4.47i·21-s + 9.48·23-s + 5.65i·27-s − 8.94i·29-s + 12·41-s + 12.7i·43-s − 9.48·47-s + 3.00·49-s + 13.4i·61-s − 3.16·63-s − 4.24i·67-s + 13.4i·69-s + ⋯
L(s)  = 1  + 0.816i·3-s − 1.19·7-s + 0.333·9-s − 0.975i·21-s + 1.97·23-s + 1.08i·27-s − 1.66i·29-s + 1.87·41-s + 1.94i·43-s − 1.38·47-s + 0.428·49-s + 1.71i·61-s − 0.398·63-s − 0.518i·67-s + 1.61i·69-s + ⋯

Functional equation

Λ(s)=(3200s/2 ΓC(s) L(s)=(−i Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(3200s/2 ΓC(s+1/2) L(s)=(−i Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 32003200    =    27⋅522^{7} \cdot 5^{2}
Sign: −i-i
Analytic conductor: 25.552125.5521
Root analytic conductor: 5.054915.05491
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ3200(1601,⋅)\chi_{3200} (1601, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3200, ( :1/2), −i)(2,\ 3200,\ (\ :1/2),\ -i)

Particular Values

L(1)L(1) ≈\approx 1.5280982451.528098245
L(12)L(\frac12) ≈\approx 1.5280982451.528098245
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
good3 1−1.41iT−3T2 1 - 1.41iT - 3T^{2}
7 1+3.16T+7T2 1 + 3.16T + 7T^{2}
11 1−11T2 1 - 11T^{2}
13 1−13T2 1 - 13T^{2}
17 1+17T2 1 + 17T^{2}
19 1−19T2 1 - 19T^{2}
23 1−9.48T+23T2 1 - 9.48T + 23T^{2}
29 1+8.94iT−29T2 1 + 8.94iT - 29T^{2}
31 1+31T2 1 + 31T^{2}
37 1−37T2 1 - 37T^{2}
41 1−12T+41T2 1 - 12T + 41T^{2}
43 1−12.7iT−43T2 1 - 12.7iT - 43T^{2}
47 1+9.48T+47T2 1 + 9.48T + 47T^{2}
53 1−53T2 1 - 53T^{2}
59 1−59T2 1 - 59T^{2}
61 1−13.4iT−61T2 1 - 13.4iT - 61T^{2}
67 1+4.24iT−67T2 1 + 4.24iT - 67T^{2}
71 1+71T2 1 + 71T^{2}
73 1+73T2 1 + 73T^{2}
79 1+79T2 1 + 79T^{2}
83 1−15.5iT−83T2 1 - 15.5iT - 83T^{2}
89 1−6T+89T2 1 - 6T + 89T^{2}
97 1+97T2 1 + 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

−9.267734746584264247516021152531, −8.145784026876213278835904311687, −7.29132315930635251732461621340, −6.56369403976058871638616933801, −5.86197909919710251196310555134, −4.85704260264121078331203349494, −4.20799995469679379846350094327, −3.32575124432422205203599727838, −2.61432698199595849931782372763, −1.00821793580506145359155993651, 0.57913438628799109247460072118, 1.68515344867657993320041582016, 2.85244636023037056017606456719, 3.53110413824044632371810507149, 4.65937574001803619265214304332, 5.55258207921656896071548240671, 6.48799290492837190035818987181, 6.96350641164489080670922143511, 7.47046846487800291775187937119, 8.515398992535737671923890554448

Graph of the ZZ-function along the critical line