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Properties

Label 2-30-15.14-c2-0-3
Degree 22
Conductor 3030
Sign 0.934+0.355i0.934 + 0.355i
Analytic cond. 0.8174400.817440
Root an. cond. 0.9041240.904124
Motivic weight 22
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.41·2-s + (−0.707 − 2.91i)3-s + 2.00·4-s + (−2.82 + 4.12i)5-s + (−1.00 − 4.12i)6-s + 5.83i·7-s + 2.82·8-s + (−8 + 4.12i)9-s + (−4.00 + 5.83i)10-s − 16.4i·11-s + (−1.41 − 5.83i)12-s + 8.24i·14-s + (14.0 + 5.33i)15-s + 4.00·16-s − 11.3·17-s + (−11.3 + 5.83i)18-s + ⋯
L(s)  = 1  + 0.707·2-s + (−0.235 − 0.971i)3-s + 0.500·4-s + (−0.565 + 0.824i)5-s + (−0.166 − 0.687i)6-s + 0.832i·7-s + 0.353·8-s + (−0.888 + 0.458i)9-s + (−0.400 + 0.583i)10-s − 1.49i·11-s + (−0.117 − 0.485i)12-s + 0.589i·14-s + (0.934 + 0.355i)15-s + 0.250·16-s − 0.665·17-s + (−0.628 + 0.323i)18-s + ⋯

Functional equation

Λ(s)=(30s/2 ΓC(s) L(s)=((0.934+0.355i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 30 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.934 + 0.355i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(30s/2 ΓC(s+1) L(s)=((0.934+0.355i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 30 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.934 + 0.355i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 3030    =    2⋅3⋅52 \cdot 3 \cdot 5
Sign: 0.934+0.355i0.934 + 0.355i
Analytic conductor: 0.8174400.817440
Root analytic conductor: 0.9041240.904124
Motivic weight: 22
Rational: no
Arithmetic: yes
Character: χ30(29,⋅)\chi_{30} (29, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 30, ( :1), 0.934+0.355i)(2,\ 30,\ (\ :1),\ 0.934 + 0.355i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 1.19370−0.219267i1.19370 - 0.219267i
L(12)L(\frac12) ≈\approx 1.19370−0.219267i1.19370 - 0.219267i
L(2)L(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 1−1.41T 1 - 1.41T
3 1+(0.707+2.91i)T 1 + (0.707 + 2.91i)T
5 1+(2.82−4.12i)T 1 + (2.82 - 4.12i)T
good7 1−5.83iT−49T2 1 - 5.83iT - 49T^{2}
11 1+16.4iT−121T2 1 + 16.4iT - 121T^{2}
13 1−169T2 1 - 169T^{2}
17 1+11.3T+289T2 1 + 11.3T + 289T^{2}
19 1−12T+361T2 1 - 12T + 361T^{2}
23 1−24.0T+529T2 1 - 24.0T + 529T^{2}
29 1−841T2 1 - 841T^{2}
31 1+32T+961T2 1 + 32T + 961T^{2}
37 1−23.3iT−1.36e3T2 1 - 23.3iT - 1.36e3T^{2}
41 1−57.7iT−1.68e3T2 1 - 57.7iT - 1.68e3T^{2}
43 1+40.8iT−1.84e3T2 1 + 40.8iT - 1.84e3T^{2}
47 1+35.3T+2.20e3T2 1 + 35.3T + 2.20e3T^{2}
53 1−67.8T+2.80e3T2 1 - 67.8T + 2.80e3T^{2}
59 1+16.4iT−3.48e3T2 1 + 16.4iT - 3.48e3T^{2}
61 1+16T+3.72e3T2 1 + 16T + 3.72e3T^{2}
67 1+5.83iT−4.48e3T2 1 + 5.83iT - 4.48e3T^{2}
71 1−5.04e3T2 1 - 5.04e3T^{2}
73 1−116.iT−5.32e3T2 1 - 116. iT - 5.32e3T^{2}
79 1+72T+6.24e3T2 1 + 72T + 6.24e3T^{2}
83 1+43.8T+6.88e3T2 1 + 43.8T + 6.88e3T^{2}
89 1−65.9iT−7.92e3T2 1 - 65.9iT - 7.92e3T^{2}
97 1+163.iT−9.40e3T2 1 + 163. iT - 9.40e3T^{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

−16.58882178716826748786556648566, −15.33053636789295574531251394776, −14.16663333127744875340328517364, −13.08710303284183496334975798865, −11.71983661285254769934238067595, −11.06588862538936461485155837294, −8.466402456698073833816059007423, −6.95497593315290938568200687329, −5.69296378855426107335447294381, −2.97760111814052126250815104749, 4.01457074796258314465551376197, 5.06168367233999760498139805777, 7.28251789770048511945899769940, 9.238455499738404659373897359660, 10.67299930059362654735356988374, 11.93040687964868699036428154939, 13.12872899599409982934838582189, 14.67942219432902335714473151718, 15.61919684673188299507500018522, 16.60555277562872714669791982688

Graph of the ZZ-function along the critical line