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Properties

Label 2-165-15.8-c1-0-9
Degree 22
Conductor 165165
Sign 0.314−0.949i0.314 - 0.949i
Analytic cond. 1.317531.31753
Root an. cond. 1.147831.14783
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  + (1.31 + 1.31i)2-s + (1.52 + 0.825i)3-s + 1.47i·4-s + (−1.49 − 1.66i)5-s + (0.918 + 3.09i)6-s + (−2.09 + 2.09i)7-s + (0.695 − 0.695i)8-s + (1.63 + 2.51i)9-s + (0.228 − 4.16i)10-s − i·11-s + (−1.21 + 2.24i)12-s + (0.161 + 0.161i)13-s − 5.51·14-s + (−0.897 − 3.76i)15-s + 4.77·16-s + (−3.37 − 3.37i)17-s + ⋯
L(s)  = 1  + (0.931 + 0.931i)2-s + (0.879 + 0.476i)3-s + 0.735i·4-s + (−0.667 − 0.744i)5-s + (0.375 + 1.26i)6-s + (−0.791 + 0.791i)7-s + (0.245 − 0.245i)8-s + (0.545 + 0.837i)9-s + (0.0721 − 1.31i)10-s − 0.301i·11-s + (−0.350 + 0.647i)12-s + (0.0448 + 0.0448i)13-s − 1.47·14-s + (−0.231 − 0.972i)15-s + 1.19·16-s + (−0.818 − 0.818i)17-s + ⋯

Functional equation

Λ(s)=(165s/2 ΓC(s) L(s)=((0.314−0.949i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.314 - 0.949i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(165s/2 ΓC(s+1/2) L(s)=((0.314−0.949i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.314 - 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 165165    =    3⋅5⋅113 \cdot 5 \cdot 11
Sign: 0.314−0.949i0.314 - 0.949i
Analytic conductor: 1.317531.31753
Root analytic conductor: 1.147831.14783
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ165(23,⋅)\chi_{165} (23, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 165, ( :1/2), 0.314−0.949i)(2,\ 165,\ (\ :1/2),\ 0.314 - 0.949i)

Particular Values

L(1)L(1) ≈\approx 1.60469+1.15914i1.60469 + 1.15914i
L(12)L(\frac12) ≈\approx 1.60469+1.15914i1.60469 + 1.15914i
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)
bad3 1+(−1.52−0.825i)T 1 + (-1.52 - 0.825i)T
5 1+(1.49+1.66i)T 1 + (1.49 + 1.66i)T
11 1+iT 1 + iT
good2 1+(−1.31−1.31i)T+2iT2 1 + (-1.31 - 1.31i)T + 2iT^{2}
7 1+(2.09−2.09i)T−7iT2 1 + (2.09 - 2.09i)T - 7iT^{2}
13 1+(−0.161−0.161i)T+13iT2 1 + (-0.161 - 0.161i)T + 13iT^{2}
17 1+(3.37+3.37i)T+17iT2 1 + (3.37 + 3.37i)T + 17iT^{2}
19 1+8.52iT−19T2 1 + 8.52iT - 19T^{2}
23 1+(3.15−3.15i)T−23iT2 1 + (3.15 - 3.15i)T - 23iT^{2}
29 1+3.01T+29T2 1 + 3.01T + 29T^{2}
31 1−5.01T+31T2 1 - 5.01T + 31T^{2}
37 1+(4.74−4.74i)T−37iT2 1 + (4.74 - 4.74i)T - 37iT^{2}
41 1−4.49iT−41T2 1 - 4.49iT - 41T^{2}
43 1+(−0.336−0.336i)T+43iT2 1 + (-0.336 - 0.336i)T + 43iT^{2}
47 1+(−6.15−6.15i)T+47iT2 1 + (-6.15 - 6.15i)T + 47iT^{2}
53 1+(1.62−1.62i)T−53iT2 1 + (1.62 - 1.62i)T - 53iT^{2}
59 1+5.16T+59T2 1 + 5.16T + 59T^{2}
61 1+3.39T+61T2 1 + 3.39T + 61T^{2}
67 1+(−9.74+9.74i)T−67iT2 1 + (-9.74 + 9.74i)T - 67iT^{2}
71 1+2.23iT−71T2 1 + 2.23iT - 71T^{2}
73 1+(−5.43−5.43i)T+73iT2 1 + (-5.43 - 5.43i)T + 73iT^{2}
79 1−5.93iT−79T2 1 - 5.93iT - 79T^{2}
83 1+(0.365−0.365i)T−83iT2 1 + (0.365 - 0.365i)T - 83iT^{2}
89 1−3.70T+89T2 1 - 3.70T + 89T^{2}
97 1+(−4.94+4.94i)T−97iT2 1 + (-4.94 + 4.94i)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

−13.37268651832860239604878079920, −12.48378788126502144176055814525, −11.21787802188179151026570469180, −9.579644690982394268610525121062, −8.888956750016994977918621365263, −7.74700631008119327928291997558, −6.63119018719156958129285070918, −5.19841871877601775646794394246, −4.34915768499635624026497800675, −3.01571106081857332206658906678, 2.16937882130937131302986705756, 3.61532460070629492148576233819, 4.00689206740429586353380623628, 6.28085266209515157041980337691, 7.41338166074995238490754695358, 8.349497971493801741539148197497, 10.07420012731094373553596468781, 10.64853450861240751759268999093, 12.05421289106303629159241090429, 12.57673104422850887135674256528

Graph of the ZZ-function along the critical line