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Properties

Label 2-3330-185.184-c1-0-83
Degree 22
Conductor 33303330
Sign 0.289+0.957i0.289 + 0.957i
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  + 2-s + 4-s + (0.718 − 2.11i)5-s − 2.74i·7-s + 8-s + (0.718 − 2.11i)10-s + 0.572·11-s + 3.04·13-s − 2.74i·14-s + 16-s + 4.95·17-s + 3.74i·19-s + (0.718 − 2.11i)20-s + 0.572·22-s + 2.98·23-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s + (0.321 − 0.946i)5-s − 1.03i·7-s + 0.353·8-s + (0.227 − 0.669i)10-s + 0.172·11-s + 0.843·13-s − 0.734i·14-s + 0.250·16-s + 1.20·17-s + 0.859i·19-s + (0.160 − 0.473i)20-s + 0.122·22-s + 0.621·23-s + ⋯

Functional equation

Λ(s)=(3330s/2 ΓC(s) L(s)=((0.289+0.957i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.289 + 0.957i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(3330s/2 ΓC(s+1/2) L(s)=((0.289+0.957i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.289 + 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 33303330    =    2⋅32⋅5⋅372 \cdot 3^{2} \cdot 5 \cdot 37
Sign: 0.289+0.957i0.289 + 0.957i
Analytic conductor: 26.590126.5901
Root analytic conductor: 5.156565.15656
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ3330(739,⋅)\chi_{3330} (739, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3330, ( :1/2), 0.289+0.957i)(2,\ 3330,\ (\ :1/2),\ 0.289 + 0.957i)

Particular Values

L(1)L(1) ≈\approx 3.4906985503.490698550
L(12)L(\frac12) ≈\approx 3.4906985503.490698550
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−T 1 - T
3 1 1
5 1+(−0.718+2.11i)T 1 + (-0.718 + 2.11i)T
37 1+(4.94−3.53i)T 1 + (4.94 - 3.53i)T
good7 1+2.74iT−7T2 1 + 2.74iT - 7T^{2}
11 1−0.572T+11T2 1 - 0.572T + 11T^{2}
13 1−3.04T+13T2 1 - 3.04T + 13T^{2}
17 1−4.95T+17T2 1 - 4.95T + 17T^{2}
19 1−3.74iT−19T2 1 - 3.74iT - 19T^{2}
23 1−2.98T+23T2 1 - 2.98T + 23T^{2}
29 1+7.36iT−29T2 1 + 7.36iT - 29T^{2}
31 1−1.12iT−31T2 1 - 1.12iT - 31T^{2}
41 1−11.7T+41T2 1 - 11.7T + 41T^{2}
43 1+6.91T+43T2 1 + 6.91T + 43T^{2}
47 1−0.776iT−47T2 1 - 0.776iT - 47T^{2}
53 1−3.55iT−53T2 1 - 3.55iT - 53T^{2}
59 1−9.27iT−59T2 1 - 9.27iT - 59T^{2}
61 1+11.2iT−61T2 1 + 11.2iT - 61T^{2}
67 1+9.09iT−67T2 1 + 9.09iT - 67T^{2}
71 1+6.26T+71T2 1 + 6.26T + 71T^{2}
73 1−7.83iT−73T2 1 - 7.83iT - 73T^{2}
79 1+0.716iT−79T2 1 + 0.716iT - 79T^{2}
83 1+2.01iT−83T2 1 + 2.01iT - 83T^{2}
89 1+8.33iT−89T2 1 + 8.33iT - 89T^{2}
97 1+15.6T+97T2 1 + 15.6T + 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

−8.264859269833452303237608298946, −7.79505286448097511747368039942, −6.89070892309519478572400895144, −5.99306611675195852211481026316, −5.49762924202322667823512048436, −4.48602745611107249496946778235, −3.95474764196877050545223900457, −3.09339134936924640505409214511, −1.66070410974902824224260270431, −0.907453490577889856936793002738, 1.42378023479272144170398235949, 2.55222070844939916695117796997, 3.15438840133752145800687142508, 3.96665349263200238679574964801, 5.27034767839467550612296109756, 5.60654111987243308876808260225, 6.48899777834118945841817242780, 7.04313529455665697439542784816, 7.910044111171461922715788720045, 8.866027324939333866094000071603

Graph of the ZZ-function along the critical line