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Properties

Label 2-3042-13.12-c1-0-16
Degree 22
Conductor 30423042
Sign −0.554−0.832i-0.554 - 0.832i
Analytic cond. 24.290424.2904
Root an. cond. 4.928534.92853
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  + i·2-s − 4-s + 2i·5-s − 2i·7-s − i·8-s − 2·10-s − 4i·11-s + 2·14-s + 16-s + 6i·19-s − 2i·20-s + 4·22-s − 4·23-s + 25-s + 2i·28-s − 8·29-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.5·4-s + 0.894i·5-s − 0.755i·7-s − 0.353i·8-s − 0.632·10-s − 1.20i·11-s + 0.534·14-s + 0.250·16-s + 1.37i·19-s − 0.447i·20-s + 0.852·22-s − 0.834·23-s + 0.200·25-s + 0.377i·28-s − 1.48·29-s + ⋯

Functional equation

Λ(s)=(3042s/2 ΓC(s) L(s)=((−0.554−0.832i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 3042 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.554 - 0.832i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(3042s/2 ΓC(s+1/2) L(s)=((−0.554−0.832i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3042 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.554 - 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 30423042    =    2⋅32⋅1322 \cdot 3^{2} \cdot 13^{2}
Sign: −0.554−0.832i-0.554 - 0.832i
Analytic conductor: 24.290424.2904
Root analytic conductor: 4.928534.92853
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ3042(1351,⋅)\chi_{3042} (1351, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3042, ( :1/2), −0.554−0.832i)(2,\ 3042,\ (\ :1/2),\ -0.554 - 0.832i)

Particular Values

L(1)L(1) ≈\approx 1.3284099871.328409987
L(12)L(\frac12) ≈\approx 1.3284099871.328409987
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−iT 1 - iT
3 1 1
13 1 1
good5 1−2iT−5T2 1 - 2iT - 5T^{2}
7 1+2iT−7T2 1 + 2iT - 7T^{2}
11 1+4iT−11T2 1 + 4iT - 11T^{2}
17 1+17T2 1 + 17T^{2}
19 1−6iT−19T2 1 - 6iT - 19T^{2}
23 1+4T+23T2 1 + 4T + 23T^{2}
29 1+8T+29T2 1 + 8T + 29T^{2}
31 1−2iT−31T2 1 - 2iT - 31T^{2}
37 1−6iT−37T2 1 - 6iT - 37T^{2}
41 1+6iT−41T2 1 + 6iT - 41T^{2}
43 1−8T+43T2 1 - 8T + 43T^{2}
47 1−8iT−47T2 1 - 8iT - 47T^{2}
53 1−12T+53T2 1 - 12T + 53T^{2}
59 1−4iT−59T2 1 - 4iT - 59T^{2}
61 1−10T+61T2 1 - 10T + 61T^{2}
67 1−2iT−67T2 1 - 2iT - 67T^{2}
71 1−16iT−71T2 1 - 16iT - 71T^{2}
73 1−14iT−73T2 1 - 14iT - 73T^{2}
79 1+4T+79T2 1 + 4T + 79T^{2}
83 1−12iT−83T2 1 - 12iT - 83T^{2}
89 1+6iT−89T2 1 + 6iT - 89T^{2}
97 1−10iT−97T2 1 - 10iT - 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

−8.736306040921971567778244045865, −8.130331027313402014022915867052, −7.35509082529192321217101711958, −6.83997034662755629319492260281, −5.88733544450536283160453938632, −5.53352626548846282311455844000, −4.04590050970486042381792713437, −3.70538416450980619881243672750, −2.57990884616682223021691345415, −1.06295113737662403086391330604, 0.47953625002604668325973577242, 1.88266447847373429979879791430, 2.45380824021374992298400825114, 3.74982092607709746520763658419, 4.56898885540164786180075272835, 5.18099701351502225675308264976, 5.95056011307064613638986078545, 7.10307472356972986103340027609, 7.80317539885099716024793577809, 8.848216874534899956306174034671

Graph of the ZZ-function along the critical line