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Properties

Label 2-3330-185.184-c1-0-63
Degree 22
Conductor 33303330
Sign 0.492+0.870i0.492 + 0.870i
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 + (−1.32 − 1.79i)5-s + 1.87i·7-s + 8-s + (−1.32 − 1.79i)10-s + 1.51·11-s − 4.78·13-s + 1.87i·14-s + 16-s + 3.99·17-s − 4.68i·19-s + (−1.32 − 1.79i)20-s + 1.51·22-s + 6.51·23-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s + (−0.594 − 0.804i)5-s + 0.707i·7-s + 0.353·8-s + (−0.420 − 0.568i)10-s + 0.455·11-s − 1.32·13-s + 0.500i·14-s + 0.250·16-s + 0.968·17-s − 1.07i·19-s + (−0.297 − 0.402i)20-s + 0.322·22-s + 1.35·23-s + ⋯

Functional equation

Λ(s)=(3330s/2 ΓC(s) L(s)=((0.492+0.870i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.492 + 0.870i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(3330s/2 ΓC(s+1/2) L(s)=((0.492+0.870i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.492 + 0.870i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 33303330    =    2⋅32⋅5⋅372 \cdot 3^{2} \cdot 5 \cdot 37
Sign: 0.492+0.870i0.492 + 0.870i
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), 0.492+0.870i)(2,\ 3330,\ (\ :1/2),\ 0.492 + 0.870i)

Particular Values

L(1)L(1) ≈\approx 2.4449325902.444932590
L(12)L(\frac12) ≈\approx 2.4449325902.444932590
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+(1.32+1.79i)T 1 + (1.32 + 1.79i)T
37 1+(6.03+0.734i)T 1 + (6.03 + 0.734i)T
good7 1−1.87iT−7T2 1 - 1.87iT - 7T^{2}
11 1−1.51T+11T2 1 - 1.51T + 11T^{2}
13 1+4.78T+13T2 1 + 4.78T + 13T^{2}
17 1−3.99T+17T2 1 - 3.99T + 17T^{2}
19 1+4.68iT−19T2 1 + 4.68iT - 19T^{2}
23 1−6.51T+23T2 1 - 6.51T + 23T^{2}
29 1−0.200iT−29T2 1 - 0.200iT - 29T^{2}
31 1+5.79iT−31T2 1 + 5.79iT - 31T^{2}
41 1+4.10T+41T2 1 + 4.10T + 41T^{2}
43 1−10.9T+43T2 1 - 10.9T + 43T^{2}
47 1+5.15iT−47T2 1 + 5.15iT - 47T^{2}
53 1−2.22iT−53T2 1 - 2.22iT - 53T^{2}
59 1+13.2iT−59T2 1 + 13.2iT - 59T^{2}
61 1+4.09iT−61T2 1 + 4.09iT - 61T^{2}
67 1−5.76iT−67T2 1 - 5.76iT - 67T^{2}
71 1−9.17T+71T2 1 - 9.17T + 71T^{2}
73 1+14.2iT−73T2 1 + 14.2iT - 73T^{2}
79 1−5.95iT−79T2 1 - 5.95iT - 79T^{2}
83 1+9.42iT−83T2 1 + 9.42iT - 83T^{2}
89 1+5.47iT−89T2 1 + 5.47iT - 89T^{2}
97 1+2.03T+97T2 1 + 2.03T + 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

−8.547602993913687112303293048084, −7.54405817729893735446215015681, −7.14128021518322835524803436213, −6.08616880899787801744027366793, −5.10368744178850519395085407866, −4.93444687345582965377602285145, −3.86273068684241691271527418487, −2.99007416605756550884116485121, −2.03712547162275820686665793848, −0.64887995651492866404113936017, 1.14583191193794361564757914542, 2.52859904882223674554327971427, 3.34518976526533772017203946404, 3.98513984547055612941260506109, 4.84894413919474982354790519948, 5.66173945148372889137419942258, 6.64353501225479396494984560712, 7.27736565578626941990260509173, 7.60001823854311689640549253310, 8.616721513367551264788163184193

Graph of the ZZ-function along the critical line