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Properties

Label 2-135-15.2-c1-0-3
Degree 22
Conductor 135135
Sign 0.725−0.688i0.725 - 0.688i
Analytic cond. 1.077981.07798
Root an. cond. 1.038251.03825
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  + (−0.323 + 0.323i)2-s + 1.79i·4-s + (1.22 − 1.87i)5-s + (1.79 + 1.79i)7-s + (−1.22 − 1.22i)8-s + (0.208 + 0.999i)10-s + 5.54i·11-s + (1.79 − 1.79i)13-s − 1.15·14-s − 2.79·16-s + (1.87 − 1.87i)17-s − 3i·19-s + (3.35 + 2.19i)20-s + (−1.79 − 1.79i)22-s + (−4.32 − 4.32i)23-s + ⋯
L(s)  = 1  + (−0.228 + 0.228i)2-s + 0.895i·4-s + (0.547 − 0.836i)5-s + (0.677 + 0.677i)7-s + (−0.433 − 0.433i)8-s + (0.0660 + 0.316i)10-s + 1.67i·11-s + (0.496 − 0.496i)13-s − 0.309·14-s − 0.697·16-s + (0.453 − 0.453i)17-s − 0.688i·19-s + (0.749 + 0.490i)20-s + (−0.381 − 0.381i)22-s + (−0.900 − 0.900i)23-s + ⋯

Functional equation

Λ(s)=(135s/2 ΓC(s) L(s)=((0.725−0.688i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.725 - 0.688i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(135s/2 ΓC(s+1/2) L(s)=((0.725−0.688i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.725 - 0.688i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 135135    =    33⋅53^{3} \cdot 5
Sign: 0.725−0.688i0.725 - 0.688i
Analytic conductor: 1.077981.07798
Root analytic conductor: 1.038251.03825
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ135(107,⋅)\chi_{135} (107, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 135, ( :1/2), 0.725−0.688i)(2,\ 135,\ (\ :1/2),\ 0.725 - 0.688i)

Particular Values

L(1)L(1) ≈\approx 1.00486+0.400969i1.00486 + 0.400969i
L(12)L(\frac12) ≈\approx 1.00486+0.400969i1.00486 + 0.400969i
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
5 1+(−1.22+1.87i)T 1 + (-1.22 + 1.87i)T
good2 1+(0.323−0.323i)T−2iT2 1 + (0.323 - 0.323i)T - 2iT^{2}
7 1+(−1.79−1.79i)T+7iT2 1 + (-1.79 - 1.79i)T + 7iT^{2}
11 1−5.54iT−11T2 1 - 5.54iT - 11T^{2}
13 1+(−1.79+1.79i)T−13iT2 1 + (-1.79 + 1.79i)T - 13iT^{2}
17 1+(−1.87+1.87i)T−17iT2 1 + (-1.87 + 1.87i)T - 17iT^{2}
19 1+3iT−19T2 1 + 3iT - 19T^{2}
23 1+(4.32+4.32i)T+23iT2 1 + (4.32 + 4.32i)T + 23iT^{2}
29 1+4.38T+29T2 1 + 4.38T + 29T^{2}
31 1+T+31T2 1 + T + 31T^{2}
37 1+(5+5i)T+37iT2 1 + (5 + 5i)T + 37iT^{2}
41 1−5.54iT−41T2 1 - 5.54iT - 41T^{2}
43 1+(−3.20+3.20i)T−43iT2 1 + (-3.20 + 3.20i)T - 43iT^{2}
47 1+(−0.646+0.646i)T−47iT2 1 + (-0.646 + 0.646i)T - 47iT^{2}
53 1+(−0.0674−0.0674i)T+53iT2 1 + (-0.0674 - 0.0674i)T + 53iT^{2}
59 1−4.38T+59T2 1 - 4.38T + 59T^{2}
61 1+T+61T2 1 + T + 61T^{2}
67 1+(−8.58−8.58i)T+67iT2 1 + (-8.58 - 8.58i)T + 67iT^{2}
71 1−5.54iT−71T2 1 - 5.54iT - 71T^{2}
73 1+(10.3−10.3i)T−73iT2 1 + (10.3 - 10.3i)T - 73iT^{2}
79 1−10.5iT−79T2 1 - 10.5iT - 79T^{2}
83 1+(8.70+8.70i)T+83iT2 1 + (8.70 + 8.70i)T + 83iT^{2}
89 1−16.6T+89T2 1 - 16.6T + 89T^{2}
97 1+(3.58+3.58i)T+97iT2 1 + (3.58 + 3.58i)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

−13.03854446736934493498721445837, −12.46846489737220114243643761998, −11.61609993281928945235842966793, −9.997577919320859843866702623095, −9.008138393135755701146257930127, −8.181330501512705467690782992552, −7.07657552102532176137596332465, −5.49023505411011861881754541121, −4.31495365375244826910554547607, −2.23795266877774120938677657403, 1.61713194195920318200066816582, 3.59922831176414174465889338660, 5.56147505710240121737893587048, 6.31806941393255689084706515881, 7.83470481704393807457943601585, 9.102594990326131027285332293248, 10.28138605531598066743605464214, 10.90185203004764225257335910539, 11.64145880607481508308750588512, 13.61489495128882556220892525395

Graph of the ZZ-function along the critical line