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Properties

Label 2-30e2-20.7-c1-0-28
Degree 22
Conductor 900900
Sign −0.630+0.775i-0.630 + 0.775i
Analytic cond. 7.186537.18653
Root an. cond. 2.680772.68077
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  + (0.0912 − 1.41i)2-s + (−1.98 − 0.257i)4-s + (1.86 − 1.86i)7-s + (−0.544 + 2.77i)8-s + 0.728i·11-s + (3.12 − 3.12i)13-s + (−2.46 − 2.80i)14-s + (3.86 + 1.02i)16-s + (1.12 + 1.12i)17-s + 3.73·19-s + (1.02 + 0.0664i)22-s + (−5.83 − 5.83i)23-s + (−4.12 − 4.69i)26-s + (−4.18 + 3.22i)28-s + 2.64i·29-s + ⋯
L(s)  = 1  + (0.0645 − 0.997i)2-s + (−0.991 − 0.128i)4-s + (0.705 − 0.705i)7-s + (−0.192 + 0.981i)8-s + 0.219i·11-s + (0.866 − 0.866i)13-s + (−0.658 − 0.749i)14-s + (0.966 + 0.255i)16-s + (0.272 + 0.272i)17-s + 0.856·19-s + (0.219 + 0.0141i)22-s + (−1.21 − 1.21i)23-s + (−0.808 − 0.920i)26-s + (−0.790 + 0.608i)28-s + 0.490i·29-s + ⋯

Functional equation

Λ(s)=(900s/2 ΓC(s) L(s)=((−0.630+0.775i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.630 + 0.775i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(900s/2 ΓC(s+1/2) L(s)=((−0.630+0.775i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.630 + 0.775i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 900900    =    22⋅32⋅522^{2} \cdot 3^{2} \cdot 5^{2}
Sign: −0.630+0.775i-0.630 + 0.775i
Analytic conductor: 7.186537.18653
Root analytic conductor: 2.680772.68077
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ900(307,⋅)\chi_{900} (307, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 900, ( :1/2), −0.630+0.775i)(2,\ 900,\ (\ :1/2),\ -0.630 + 0.775i)

Particular Values

L(1)L(1) ≈\approx 0.649389−1.36505i0.649389 - 1.36505i
L(12)L(\frac12) ≈\approx 0.649389−1.36505i0.649389 - 1.36505i
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+(−0.0912+1.41i)T 1 + (-0.0912 + 1.41i)T
3 1 1
5 1 1
good7 1+(−1.86+1.86i)T−7iT2 1 + (-1.86 + 1.86i)T - 7iT^{2}
11 1−0.728iT−11T2 1 - 0.728iT - 11T^{2}
13 1+(−3.12+3.12i)T−13iT2 1 + (-3.12 + 3.12i)T - 13iT^{2}
17 1+(−1.12−1.12i)T+17iT2 1 + (-1.12 - 1.12i)T + 17iT^{2}
19 1−3.73T+19T2 1 - 3.73T + 19T^{2}
23 1+(5.83+5.83i)T+23iT2 1 + (5.83 + 5.83i)T + 23iT^{2}
29 1−2.64iT−29T2 1 - 2.64iT - 29T^{2}
31 1+6.01iT−31T2 1 + 6.01iT - 31T^{2}
37 1+(3.12+3.12i)T+37iT2 1 + (3.12 + 3.12i)T + 37iT^{2}
41 1−4.24T+41T2 1 - 4.24T + 41T^{2}
43 1+(5.10+5.10i)T+43iT2 1 + (5.10 + 5.10i)T + 43iT^{2}
47 1+(−2.09+2.09i)T−47iT2 1 + (-2.09 + 2.09i)T - 47iT^{2}
53 1+(−0.484+0.484i)T−53iT2 1 + (-0.484 + 0.484i)T - 53iT^{2}
59 1+4.92T+59T2 1 + 4.92T + 59T^{2}
61 1−2.31T+61T2 1 - 2.31T + 61T^{2}
67 1+(−5.10+5.10i)T−67iT2 1 + (-5.10 + 5.10i)T - 67iT^{2}
71 1+13.1iT−71T2 1 + 13.1iT - 71T^{2}
73 1+(3.96−3.96i)T−73iT2 1 + (3.96 - 3.96i)T - 73iT^{2}
79 1+7.11T+79T2 1 + 7.11T + 79T^{2}
83 1+(−3.55−3.55i)T+83iT2 1 + (-3.55 - 3.55i)T + 83iT^{2}
89 1−1.03iT−89T2 1 - 1.03iT - 89T^{2}
97 1+(−12.5−12.5i)T+97iT2 1 + (-12.5 - 12.5i)T + 97iT^{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

−10.16362498585803973166751835264, −9.085231477389253999679052987318, −8.187728452726754815942955934293, −7.59770542747819315788189140428, −6.13009330122418530002101818622, −5.19073321394700387419946595816, −4.20288612528679100498303044975, −3.40488182811004410832130632834, −2.04441903337989233496878090304, −0.797338722993198515233733876793, 1.51151734728592028831414450572, 3.30092923139598956347071398899, 4.36992275868305774741173123142, 5.37697203694032111347849946711, 6.00060169103528016177791257832, 7.01456429584995192934994143230, 7.918863362942762123258545870473, 8.581607048872550030586544591517, 9.330409892996151581880536377006, 10.13687584049148305250368386981

Graph of the ZZ-function along the critical line