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Properties

Label 2-30e2-100.23-c1-0-26
Degree 22
Conductor 900900
Sign 0.942+0.334i0.942 + 0.334i
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  + (0.213 − 1.39i)2-s + (−1.90 − 0.595i)4-s + (1.90 + 1.17i)5-s + (2.16 + 2.16i)7-s + (−1.23 + 2.54i)8-s + (2.04 − 2.40i)10-s + (−3.61 − 4.97i)11-s + (0.749 + 4.72i)13-s + (3.48 − 2.56i)14-s + (3.29 + 2.27i)16-s + (3.51 + 1.79i)17-s + (0.678 + 2.08i)19-s + (−2.93 − 3.37i)20-s + (−7.73 + 3.99i)22-s + (−0.374 + 2.36i)23-s + ⋯
L(s)  = 1  + (0.150 − 0.988i)2-s + (−0.954 − 0.297i)4-s + (0.850 + 0.525i)5-s + (0.817 + 0.817i)7-s + (−0.438 + 0.898i)8-s + (0.647 − 0.762i)10-s + (−1.09 − 1.50i)11-s + (0.207 + 1.31i)13-s + (0.931 − 0.685i)14-s + (0.822 + 0.568i)16-s + (0.852 + 0.434i)17-s + (0.155 + 0.479i)19-s + (−0.655 − 0.754i)20-s + (−1.64 + 0.852i)22-s + (−0.0781 + 0.493i)23-s + ⋯

Functional equation

Λ(s)=(900s/2 ΓC(s) L(s)=((0.942+0.334i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.942 + 0.334i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(900s/2 ΓC(s+1/2) L(s)=((0.942+0.334i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.942 + 0.334i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 900900    =    22⋅32⋅522^{2} \cdot 3^{2} \cdot 5^{2}
Sign: 0.942+0.334i0.942 + 0.334i
Analytic conductor: 7.186537.18653
Root analytic conductor: 2.680772.68077
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ900(523,⋅)\chi_{900} (523, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 900, ( :1/2), 0.942+0.334i)(2,\ 900,\ (\ :1/2),\ 0.942 + 0.334i)

Particular Values

L(1)L(1) ≈\approx 1.81015−0.311325i1.81015 - 0.311325i
L(12)L(\frac12) ≈\approx 1.81015−0.311325i1.81015 - 0.311325i
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+(−0.213+1.39i)T 1 + (-0.213 + 1.39i)T
3 1 1
5 1+(−1.90−1.17i)T 1 + (-1.90 - 1.17i)T
good7 1+(−2.16−2.16i)T+7iT2 1 + (-2.16 - 2.16i)T + 7iT^{2}
11 1+(3.61+4.97i)T+(−3.39+10.4i)T2 1 + (3.61 + 4.97i)T + (-3.39 + 10.4i)T^{2}
13 1+(−0.749−4.72i)T+(−12.3+4.01i)T2 1 + (-0.749 - 4.72i)T + (-12.3 + 4.01i)T^{2}
17 1+(−3.51−1.79i)T+(9.99+13.7i)T2 1 + (-3.51 - 1.79i)T + (9.99 + 13.7i)T^{2}
19 1+(−0.678−2.08i)T+(−15.3+11.1i)T2 1 + (-0.678 - 2.08i)T + (-15.3 + 11.1i)T^{2}
23 1+(0.374−2.36i)T+(−21.8−7.10i)T2 1 + (0.374 - 2.36i)T + (-21.8 - 7.10i)T^{2}
29 1+(1.48+0.480i)T+(23.4+17.0i)T2 1 + (1.48 + 0.480i)T + (23.4 + 17.0i)T^{2}
31 1+(−8.29+2.69i)T+(25.0−18.2i)T2 1 + (-8.29 + 2.69i)T + (25.0 - 18.2i)T^{2}
37 1+(−4.07+0.644i)T+(35.1−11.4i)T2 1 + (-4.07 + 0.644i)T + (35.1 - 11.4i)T^{2}
41 1+(−5.32−3.86i)T+(12.6+38.9i)T2 1 + (-5.32 - 3.86i)T + (12.6 + 38.9i)T^{2}
43 1+(8.72−8.72i)T−43iT2 1 + (8.72 - 8.72i)T - 43iT^{2}
47 1+(−3.47+1.77i)T+(27.6−38.0i)T2 1 + (-3.47 + 1.77i)T + (27.6 - 38.0i)T^{2}
53 1+(−5.54+2.82i)T+(31.1−42.8i)T2 1 + (-5.54 + 2.82i)T + (31.1 - 42.8i)T^{2}
59 1+(7.51+5.46i)T+(18.2+56.1i)T2 1 + (7.51 + 5.46i)T + (18.2 + 56.1i)T^{2}
61 1+(0.715−0.519i)T+(18.8−58.0i)T2 1 + (0.715 - 0.519i)T + (18.8 - 58.0i)T^{2}
67 1+(−1.52+2.98i)T+(−39.3−54.2i)T2 1 + (-1.52 + 2.98i)T + (-39.3 - 54.2i)T^{2}
71 1+(5.96+1.93i)T+(57.4+41.7i)T2 1 + (5.96 + 1.93i)T + (57.4 + 41.7i)T^{2}
73 1+(−0.906−0.143i)T+(69.4+22.5i)T2 1 + (-0.906 - 0.143i)T + (69.4 + 22.5i)T^{2}
79 1+(−2.98+9.17i)T+(−63.9−46.4i)T2 1 + (-2.98 + 9.17i)T + (-63.9 - 46.4i)T^{2}
83 1+(−6.49−3.31i)T+(48.7+67.1i)T2 1 + (-6.49 - 3.31i)T + (48.7 + 67.1i)T^{2}
89 1+(0.440+0.606i)T+(−27.5+84.6i)T2 1 + (0.440 + 0.606i)T + (-27.5 + 84.6i)T^{2}
97 1+(7.49+14.7i)T+(−57.0+78.4i)T2 1 + (7.49 + 14.7i)T + (-57.0 + 78.4i)T^{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

−10.11802103674572688480552728599, −9.453670277433350561674761333761, −8.500509693527912398790480486167, −7.910909086021980594454525977230, −6.15217810417298522146555442964, −5.69373328375882962839800496002, −4.73830314057303887329945314908, −3.36989837021267752598167538843, −2.48709498532910147672588618672, −1.45883775908046964418610106721, 0.960439501459766255171930136021, 2.69212998620400082975752705063, 4.31951277903398438477799639001, 5.05016288608265463282294478621, 5.58551003484206922869247025839, 6.84936823697857497011602515615, 7.68180491952461002725909065479, 8.154328875528721060265077035300, 9.246210698966387276688722642291, 10.22489597104581188439777942464

Graph of the ZZ-function along the critical line