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Properties

Label 2-3808-952.237-c0-0-4
Degree 22
Conductor 38083808
Sign 0.309−0.951i0.309 - 0.951i
Analytic cond. 1.900431.90043
Root an. cond. 1.378561.37856
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.90i·3-s − 1.17i·5-s + 7-s − 2.61·9-s + 2.23·15-s + 17-s + 1.90i·21-s − 0.381·25-s − 3.07i·27-s + 1.61·31-s − 1.17i·35-s − 0.618·41-s + 1.90i·43-s + 3.07i·45-s + 49-s + ⋯
L(s)  = 1  + 1.90i·3-s − 1.17i·5-s + 7-s − 2.61·9-s + 2.23·15-s + 17-s + 1.90i·21-s − 0.381·25-s − 3.07i·27-s + 1.61·31-s − 1.17i·35-s − 0.618·41-s + 1.90i·43-s + 3.07i·45-s + 49-s + ⋯

Functional equation

Λ(s)=(3808s/2 ΓC(s) L(s)=((0.309−0.951i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3808 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.309 - 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(3808s/2 ΓC(s) L(s)=((0.309−0.951i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3808 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.309 - 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 38083808    =    25⋅7⋅172^{5} \cdot 7 \cdot 17
Sign: 0.309−0.951i0.309 - 0.951i
Analytic conductor: 1.900431.90043
Root analytic conductor: 1.378561.37856
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3808(3569,⋅)\chi_{3808} (3569, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3808, ( :0), 0.309−0.951i)(2,\ 3808,\ (\ :0),\ 0.309 - 0.951i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.4248916991.424891699
L(12)L(\frac12) ≈\approx 1.4248916991.424891699
L(1)L(1) 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
7 1−T 1 - T
17 1−T 1 - T
good3 1−1.90iT−T2 1 - 1.90iT - T^{2}
5 1+1.17iT−T2 1 + 1.17iT - T^{2}
11 1+T2 1 + T^{2}
13 1+T2 1 + T^{2}
19 1+T2 1 + T^{2}
23 1−T2 1 - T^{2}
29 1+T2 1 + T^{2}
31 1−1.61T+T2 1 - 1.61T + T^{2}
37 1+T2 1 + T^{2}
41 1+0.618T+T2 1 + 0.618T + T^{2}
43 1−1.90iT−T2 1 - 1.90iT - T^{2}
47 1−T2 1 - T^{2}
53 1−1.17iT−T2 1 - 1.17iT - T^{2}
59 1+T2 1 + T^{2}
61 1+1.90iT−T2 1 + 1.90iT - T^{2}
67 1−1.17iT−T2 1 - 1.17iT - T^{2}
71 1−T2 1 - T^{2}
73 1−0.618T+T2 1 - 0.618T + T^{2}
79 1−T2 1 - T^{2}
83 1+T2 1 + T^{2}
89 1−T2 1 - T^{2}
97 1−1.61T+T2 1 - 1.61T + T^{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

−8.901873069103863427741912746592, −8.264139518971691248545644220403, −7.83373763752806333799851058333, −6.26134955352158490656437796660, −5.44680303014830919589535263719, −4.84430981138075124449622775563, −4.52203560917423071567810775307, −3.64511799659171553158835914152, −2.68638259072815263849897155521, −1.16836248226138173994248206264, 1.01955770609237974815582454792, 2.01237503475484045843156791844, 2.69590975403555762289568677486, 3.57189688292854495768567703790, 5.03216854716428668642060173521, 5.80113768708420282360123574107, 6.53873918865929285003444528177, 7.09562211211563206774181965664, 7.68034663816381061871127267557, 8.220008707161007166195492136074

Graph of the ZZ-function along the critical line