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Properties

Label 2-1110-37.36-c1-0-12
Degree 22
Conductor 11101110
Sign 0.986+0.164i0.986 + 0.164i
Analytic cond. 8.863398.86339
Root an. cond. 2.977142.97714
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 − 3-s − 4-s + i·5-s − i·6-s − 7-s − i·8-s + 9-s − 10-s − 11-s + 12-s − i·13-s − i·14-s − i·15-s + 16-s − 5i·17-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.577·3-s − 0.5·4-s + 0.447i·5-s − 0.408i·6-s − 0.377·7-s − 0.353i·8-s + 0.333·9-s − 0.316·10-s − 0.301·11-s + 0.288·12-s − 0.277i·13-s − 0.267i·14-s − 0.258i·15-s + 0.250·16-s − 1.21i·17-s + ⋯

Functional equation

Λ(s)=(1110s/2 ΓC(s) L(s)=((0.986+0.164i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.986 + 0.164i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1110s/2 ΓC(s+1/2) L(s)=((0.986+0.164i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.986 + 0.164i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 11101110    =    2⋅3⋅5⋅372 \cdot 3 \cdot 5 \cdot 37
Sign: 0.986+0.164i0.986 + 0.164i
Analytic conductor: 8.863398.86339
Root analytic conductor: 2.977142.97714
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1110(961,⋅)\chi_{1110} (961, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1110, ( :1/2), 0.986+0.164i)(2,\ 1110,\ (\ :1/2),\ 0.986 + 0.164i)

Particular Values

L(1)L(1) ≈\approx 0.92594286300.9259428630
L(12)L(\frac12) ≈\approx 0.92594286300.9259428630
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+T 1 + T
5 1−iT 1 - iT
37 1+(−6−i)T 1 + (-6 - i)T
good7 1+T+7T2 1 + T + 7T^{2}
11 1+T+11T2 1 + T + 11T^{2}
13 1+iT−13T2 1 + iT - 13T^{2}
17 1+5iT−17T2 1 + 5iT - 17T^{2}
19 1+3iT−19T2 1 + 3iT - 19T^{2}
23 1+3iT−23T2 1 + 3iT - 23T^{2}
29 1−10iT−29T2 1 - 10iT - 29T^{2}
31 1+2iT−31T2 1 + 2iT - 31T^{2}
41 1+2T+41T2 1 + 2T + 41T^{2}
43 1+12iT−43T2 1 + 12iT - 43T^{2}
47 1−12T+47T2 1 - 12T + 47T^{2}
53 1−11T+53T2 1 - 11T + 53T^{2}
59 1+10iT−59T2 1 + 10iT - 59T^{2}
61 1+14iT−61T2 1 + 14iT - 61T^{2}
67 1−12T+67T2 1 - 12T + 67T^{2}
71 1−6T+71T2 1 - 6T + 71T^{2}
73 1+T+73T2 1 + T + 73T^{2}
79 1−10iT−79T2 1 - 10iT - 79T^{2}
83 1+3T+83T2 1 + 3T + 83T^{2}
89 1−9iT−89T2 1 - 9iT - 89T^{2}
97 1−12iT−97T2 1 - 12iT - 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

−9.757674592322395651360084876713, −9.023074740307522681850076441493, −8.020486131892934550098755552225, −6.99416720499669562100549753595, −6.72400727434630523911025153399, −5.51968528325553463368885747218, −4.96389623615838547188196016628, −3.73114384413749356116416006896, −2.56023766661685018531833096121, −0.52536425682019317544191150121, 1.11520569429418792601494087728, 2.38677319813039334606711147299, 3.78866922353619549736702948144, 4.46116264552764782255529051814, 5.66328319385818833104268411075, 6.19429555039558421062594892004, 7.52301015595131319283184415607, 8.303894586725900257113948758249, 9.262152701564750930883455918575, 10.02592590252124287993361174379

Graph of the ZZ-function along the critical line