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Properties

Label 2-3330-37.36-c1-0-55
Degree 22
Conductor 33303330
Sign −0.164+0.986i-0.164 + 0.986i
Analytic cond. 26.590126.5901
Root an. cond. 5.156565.15656
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 + i·5-s + 7-s − i·8-s − 10-s − 3·11-s − 6i·13-s + i·14-s + 16-s + 3i·17-s + 6i·19-s − i·20-s − 3i·22-s − 6i·23-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.5·4-s + 0.447i·5-s + 0.377·7-s − 0.353i·8-s − 0.316·10-s − 0.904·11-s − 1.66i·13-s + 0.267i·14-s + 0.250·16-s + 0.727i·17-s + 1.37i·19-s − 0.223i·20-s − 0.639i·22-s − 1.25i·23-s + ⋯

Functional equation

Λ(s)=(3330s/2 ΓC(s) L(s)=((−0.164+0.986i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.164 + 0.986i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(3330s/2 ΓC(s+1/2) L(s)=((−0.164+0.986i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.164 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 33303330    =    2⋅32⋅5⋅372 \cdot 3^{2} \cdot 5 \cdot 37
Sign: −0.164+0.986i-0.164 + 0.986i
Analytic conductor: 26.590126.5901
Root analytic conductor: 5.156565.15656
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ3330(2071,⋅)\chi_{3330} (2071, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3330, ( :1/2), −0.164+0.986i)(2,\ 3330,\ (\ :1/2),\ -0.164 + 0.986i)

Particular Values

L(1)L(1) ≈\approx 0.27788021600.2778802160
L(12)L(\frac12) ≈\approx 0.27788021600.2778802160
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
5 1−iT 1 - iT
37 1+(−1+6i)T 1 + (-1 + 6i)T
good7 1−T+7T2 1 - T + 7T^{2}
11 1+3T+11T2 1 + 3T + 11T^{2}
13 1+6iT−13T2 1 + 6iT - 13T^{2}
17 1−3iT−17T2 1 - 3iT - 17T^{2}
19 1−6iT−19T2 1 - 6iT - 19T^{2}
23 1+6iT−23T2 1 + 6iT - 23T^{2}
29 1−9iT−29T2 1 - 9iT - 29T^{2}
31 1−3iT−31T2 1 - 3iT - 31T^{2}
41 1+9T+41T2 1 + 9T + 41T^{2}
43 1+9iT−43T2 1 + 9iT - 43T^{2}
47 1+47T2 1 + 47T^{2}
53 1+3T+53T2 1 + 3T + 53T^{2}
59 1−59T2 1 - 59T^{2}
61 1+9iT−61T2 1 + 9iT - 61T^{2}
67 1+14T+67T2 1 + 14T + 67T^{2}
71 1+6T+71T2 1 + 6T + 71T^{2}
73 1+2T+73T2 1 + 2T + 73T^{2}
79 1−79T2 1 - 79T^{2}
83 1+83T2 1 + 83T^{2}
89 1+12iT−89T2 1 + 12iT - 89T^{2}
97 1−3iT−97T2 1 - 3iT - 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

−8.256045665020811152954229247989, −7.75686600491650076195366328942, −7.00725728676128280281215264287, −6.09904064115191180052452910688, −5.46753985966985822598172047265, −4.84044807002322263231879041209, −3.65593859238846918569404562188, −2.97150232130048190712512345387, −1.66695384825412519553343323703, −0.083806048222460538678434630380, 1.34102026886197044130013945807, 2.28135096276193073031715815484, 3.13573136308209404159706203882, 4.42962014701178618038349445039, 4.67826242140415900847189877807, 5.61892550125665444279650634245, 6.61851589649924298723094933432, 7.49885702635880212026379367073, 8.173057741731193378673411281362, 9.029169250814739430084650571497

Graph of the ZZ-function along the critical line