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Properties

Label 2-2368-37.36-c1-0-21
Degree 22
Conductor 23682368
Sign −0.657−0.753i-0.657 - 0.753i
Analytic cond. 18.908518.9085
Root an. cond. 4.348394.34839
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  + 2.79·3-s + 3.79i·5-s − 2·7-s + 4.79·9-s − 3.79·11-s − 0.791i·13-s + 10.5i·15-s − 1.58i·17-s + 7.58i·19-s − 5.58·21-s + 0.791i·23-s − 9.37·25-s + 4.99·27-s + 0.791i·29-s + 5.37i·31-s + ⋯
L(s)  = 1  + 1.61·3-s + 1.69i·5-s − 0.755·7-s + 1.59·9-s − 1.14·11-s − 0.219i·13-s + 2.73i·15-s − 0.383i·17-s + 1.73i·19-s − 1.21·21-s + 0.164i·23-s − 1.87·25-s + 0.962·27-s + 0.146i·29-s + 0.965i·31-s + ⋯

Functional equation

Λ(s)=(2368s/2 ΓC(s) L(s)=((−0.657−0.753i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 2368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.657 - 0.753i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(2368s/2 ΓC(s+1/2) L(s)=((−0.657−0.753i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.657 - 0.753i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 23682368    =    26⋅372^{6} \cdot 37
Sign: −0.657−0.753i-0.657 - 0.753i
Analytic conductor: 18.908518.9085
Root analytic conductor: 4.348394.34839
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ2368(961,⋅)\chi_{2368} (961, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2368, ( :1/2), −0.657−0.753i)(2,\ 2368,\ (\ :1/2),\ -0.657 - 0.753i)

Particular Values

L(1)L(1) ≈\approx 2.1351628022.135162802
L(12)L(\frac12) ≈\approx 2.1351628022.135162802
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
37 1+(4+4.58i)T 1 + (4 + 4.58i)T
good3 1−2.79T+3T2 1 - 2.79T + 3T^{2}
5 1−3.79iT−5T2 1 - 3.79iT - 5T^{2}
7 1+2T+7T2 1 + 2T + 7T^{2}
11 1+3.79T+11T2 1 + 3.79T + 11T^{2}
13 1+0.791iT−13T2 1 + 0.791iT - 13T^{2}
17 1+1.58iT−17T2 1 + 1.58iT - 17T^{2}
19 1−7.58iT−19T2 1 - 7.58iT - 19T^{2}
23 1−0.791iT−23T2 1 - 0.791iT - 23T^{2}
29 1−0.791iT−29T2 1 - 0.791iT - 29T^{2}
31 1−5.37iT−31T2 1 - 5.37iT - 31T^{2}
41 1−5.20T+41T2 1 - 5.20T + 41T^{2}
43 1−6iT−43T2 1 - 6iT - 43T^{2}
47 1−1.58T+47T2 1 - 1.58T + 47T^{2}
53 1+7.58T+53T2 1 + 7.58T + 53T^{2}
59 1−7.58iT−59T2 1 - 7.58iT - 59T^{2}
61 1+8.20iT−61T2 1 + 8.20iT - 61T^{2}
67 1−7.37T+67T2 1 - 7.37T + 67T^{2}
71 1−9.16T+71T2 1 - 9.16T + 71T^{2}
73 1−9.37T+73T2 1 - 9.37T + 73T^{2}
79 1−12.7iT−79T2 1 - 12.7iT - 79T^{2}
83 1−3.16T+83T2 1 - 3.16T + 83T^{2}
89 1−6iT−89T2 1 - 6iT - 89T^{2}
97 1+4.41iT−97T2 1 + 4.41iT - 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

−9.447671436703243390174000094296, −8.222574532934674868974645967015, −7.83933476409230407025945932743, −7.12057995466516196225358823544, −6.41150368932250250680772871955, −5.42799649979192886927991131571, −3.92217155651008412186320171042, −3.25705491625013597938919224255, −2.81080279284915227828569898289, −1.96191210290417094245436604385, 0.53024619584692557849770158882, 1.96772242624668441046302056651, 2.76874914202589151159143031377, 3.74652266801868183118272949905, 4.61641554909603812503223250174, 5.28597592303877060713771939646, 6.48124805816375689959541242271, 7.52397521145450639807884286036, 8.131664731025099697561730800888, 8.718682020742154788453587652056

Graph of the ZZ-function along the critical line