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Properties

Label 2-30e2-100.91-c0-0-0
Degree 22
Conductor 900900
Sign −0.187−0.982i-0.187 - 0.982i
Analytic cond. 0.4491580.449158
Root an. cond. 0.6701920.670192
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.809 + 0.587i)2-s + (0.309 + 0.951i)4-s + (−0.309 + 0.951i)5-s + (−0.309 + 0.951i)8-s + (−0.809 + 0.587i)10-s + (−0.5 + 0.363i)13-s + (−0.809 + 0.587i)16-s + (0.5 − 1.53i)17-s − 20-s + (−0.809 − 0.587i)25-s − 0.618·26-s + (0.5 + 1.53i)29-s − 32-s + (1.30 − 0.951i)34-s + (1.30 − 0.951i)37-s + ⋯
L(s)  = 1  + (0.809 + 0.587i)2-s + (0.309 + 0.951i)4-s + (−0.309 + 0.951i)5-s + (−0.309 + 0.951i)8-s + (−0.809 + 0.587i)10-s + (−0.5 + 0.363i)13-s + (−0.809 + 0.587i)16-s + (0.5 − 1.53i)17-s − 20-s + (−0.809 − 0.587i)25-s − 0.618·26-s + (0.5 + 1.53i)29-s − 32-s + (1.30 − 0.951i)34-s + (1.30 − 0.951i)37-s + ⋯

Functional equation

Λ(s)=(900s/2 ΓC(s) L(s)=((−0.187−0.982i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.187 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(900s/2 ΓC(s) L(s)=((−0.187−0.982i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.187 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 900900    =    22⋅32⋅522^{2} \cdot 3^{2} \cdot 5^{2}
Sign: −0.187−0.982i-0.187 - 0.982i
Analytic conductor: 0.4491580.449158
Root analytic conductor: 0.6701920.670192
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ900(91,⋅)\chi_{900} (91, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 900, ( :0), −0.187−0.982i)(2,\ 900,\ (\ :0),\ -0.187 - 0.982i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.4239631611.423963161
L(12)L(\frac12) ≈\approx 1.4239631611.423963161
L(1)L(1) 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+(−0.809−0.587i)T 1 + (-0.809 - 0.587i)T
3 1 1
5 1+(0.309−0.951i)T 1 + (0.309 - 0.951i)T
good7 1−T2 1 - T^{2}
11 1+(−0.309−0.951i)T2 1 + (-0.309 - 0.951i)T^{2}
13 1+(0.5−0.363i)T+(0.309−0.951i)T2 1 + (0.5 - 0.363i)T + (0.309 - 0.951i)T^{2}
17 1+(−0.5+1.53i)T+(−0.809−0.587i)T2 1 + (-0.5 + 1.53i)T + (-0.809 - 0.587i)T^{2}
19 1+(0.809+0.587i)T2 1 + (0.809 + 0.587i)T^{2}
23 1+(−0.309−0.951i)T2 1 + (-0.309 - 0.951i)T^{2}
29 1+(−0.5−1.53i)T+(−0.809+0.587i)T2 1 + (-0.5 - 1.53i)T + (-0.809 + 0.587i)T^{2}
31 1+(0.809+0.587i)T2 1 + (0.809 + 0.587i)T^{2}
37 1+(−1.30+0.951i)T+(0.309−0.951i)T2 1 + (-1.30 + 0.951i)T + (0.309 - 0.951i)T^{2}
41 1+(−0.5+0.363i)T+(0.309−0.951i)T2 1 + (-0.5 + 0.363i)T + (0.309 - 0.951i)T^{2}
43 1−T2 1 - T^{2}
47 1+(0.809−0.587i)T2 1 + (0.809 - 0.587i)T^{2}
53 1+(0.190+0.587i)T+(−0.809+0.587i)T2 1 + (0.190 + 0.587i)T + (-0.809 + 0.587i)T^{2}
59 1+(−0.309+0.951i)T2 1 + (-0.309 + 0.951i)T^{2}
61 1+(0.5+0.363i)T+(0.309+0.951i)T2 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2}
67 1+(0.809+0.587i)T2 1 + (0.809 + 0.587i)T^{2}
71 1+(0.809−0.587i)T2 1 + (0.809 - 0.587i)T^{2}
73 1+(0.5+0.363i)T+(0.309+0.951i)T2 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2}
79 1+(0.809−0.587i)T2 1 + (0.809 - 0.587i)T^{2}
83 1+(0.809+0.587i)T2 1 + (0.809 + 0.587i)T^{2}
89 1+(1.30+0.951i)T+(0.309+0.951i)T2 1 + (1.30 + 0.951i)T + (0.309 + 0.951i)T^{2}
97 1+(0.5+1.53i)T+(−0.809+0.587i)T2 1 + (0.5 + 1.53i)T + (-0.809 + 0.587i)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

−10.76486800999752715139386487573, −9.683651154086042175510409792962, −8.735601538043212685785742143112, −7.51037078062743494332919133313, −7.24157482103488924726725352968, −6.29984054906616827132096526159, −5.30764636196956021626146305528, −4.37427395437747443119762081843, −3.28030407126397008424340129123, −2.47470684055183168209442359936, 1.23149490184701595948393562657, 2.61240193002872695376815841275, 3.91436252679634624073741527626, 4.55234985015706056675924116951, 5.59659764838152895583636657785, 6.26717098598768207066738525179, 7.63171877349717513785500929750, 8.385713789739642149276021526797, 9.500281353636004488824856649603, 10.13110540383176579835939850246

Graph of the ZZ-function along the critical line