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Properties

Label 2-275-55.14-c1-0-14
Degree 22
Conductor 275275
Sign −0.999−0.00747i-0.999 - 0.00747i
Analytic cond. 2.195882.19588
Root an. cond. 1.481851.48185
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  + (−0.796 − 1.09i)2-s + (0.547 − 0.177i)3-s + (0.0501 − 0.154i)4-s + (−0.631 − 0.458i)6-s + (−3.47 − 1.12i)7-s + (−2.78 + 0.905i)8-s + (−2.15 + 1.56i)9-s + (0.490 − 3.28i)11-s − 0.0933i·12-s + (−1.66 − 2.29i)13-s + (1.52 + 4.70i)14-s + (2.95 + 2.14i)16-s + (2.17 − 2.98i)17-s + (3.44 + 1.11i)18-s + (0.0293 + 0.0904i)19-s + ⋯
L(s)  = 1  + (−0.563 − 0.775i)2-s + (0.315 − 0.102i)3-s + (0.0250 − 0.0771i)4-s + (−0.257 − 0.187i)6-s + (−1.31 − 0.426i)7-s + (−0.985 + 0.320i)8-s + (−0.719 + 0.522i)9-s + (0.147 − 0.989i)11-s − 0.0269i·12-s + (−0.461 − 0.635i)13-s + (0.408 + 1.25i)14-s + (0.738 + 0.536i)16-s + (0.526 − 0.724i)17-s + (0.811 + 0.263i)18-s + (0.00674 + 0.0207i)19-s + ⋯

Functional equation

Λ(s)=(275s/2 ΓC(s) L(s)=((−0.999−0.00747i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.00747i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(275s/2 ΓC(s+1/2) L(s)=((−0.999−0.00747i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.00747i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 275275    =    52⋅115^{2} \cdot 11
Sign: −0.999−0.00747i-0.999 - 0.00747i
Analytic conductor: 2.195882.19588
Root analytic conductor: 1.481851.48185
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ275(124,⋅)\chi_{275} (124, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 275, ( :1/2), −0.999−0.00747i)(2,\ 275,\ (\ :1/2),\ -0.999 - 0.00747i)

Particular Values

L(1)L(1) ≈\approx 0.00220525+0.589733i0.00220525 + 0.589733i
L(12)L(\frac12) ≈\approx 0.00220525+0.589733i0.00220525 + 0.589733i
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)
bad5 1 1
11 1+(−0.490+3.28i)T 1 + (-0.490 + 3.28i)T
good2 1+(0.796+1.09i)T+(−0.618+1.90i)T2 1 + (0.796 + 1.09i)T + (-0.618 + 1.90i)T^{2}
3 1+(−0.547+0.177i)T+(2.42−1.76i)T2 1 + (-0.547 + 0.177i)T + (2.42 - 1.76i)T^{2}
7 1+(3.47+1.12i)T+(5.66+4.11i)T2 1 + (3.47 + 1.12i)T + (5.66 + 4.11i)T^{2}
13 1+(1.66+2.29i)T+(−4.01+12.3i)T2 1 + (1.66 + 2.29i)T + (-4.01 + 12.3i)T^{2}
17 1+(−2.17+2.98i)T+(−5.25−16.1i)T2 1 + (-2.17 + 2.98i)T + (-5.25 - 16.1i)T^{2}
19 1+(−0.0293−0.0904i)T+(−15.3+11.1i)T2 1 + (-0.0293 - 0.0904i)T + (-15.3 + 11.1i)T^{2}
23 1+1.16iT−23T2 1 + 1.16iT - 23T^{2}
29 1+(−2.08+6.42i)T+(−23.4−17.0i)T2 1 + (-2.08 + 6.42i)T + (-23.4 - 17.0i)T^{2}
31 1+(5.48−3.98i)T+(9.57−29.4i)T2 1 + (5.48 - 3.98i)T + (9.57 - 29.4i)T^{2}
37 1+(−9.35−3.04i)T+(29.9+21.7i)T2 1 + (-9.35 - 3.04i)T + (29.9 + 21.7i)T^{2}
41 1+(2.57+7.91i)T+(−33.1+24.0i)T2 1 + (2.57 + 7.91i)T + (-33.1 + 24.0i)T^{2}
43 1−2.96iT−43T2 1 - 2.96iT - 43T^{2}
47 1+(−2.11+0.687i)T+(38.0−27.6i)T2 1 + (-2.11 + 0.687i)T + (38.0 - 27.6i)T^{2}
53 1+(−1.75−2.42i)T+(−16.3+50.4i)T2 1 + (-1.75 - 2.42i)T + (-16.3 + 50.4i)T^{2}
59 1+(−2.62+8.09i)T+(−47.7−34.6i)T2 1 + (-2.62 + 8.09i)T + (-47.7 - 34.6i)T^{2}
61 1+(−6.86−4.98i)T+(18.8+58.0i)T2 1 + (-6.86 - 4.98i)T + (18.8 + 58.0i)T^{2}
67 1+13.4iT−67T2 1 + 13.4iT - 67T^{2}
71 1+(6.71+4.88i)T+(21.9+67.5i)T2 1 + (6.71 + 4.88i)T + (21.9 + 67.5i)T^{2}
73 1+(−1.25−0.407i)T+(59.0+42.9i)T2 1 + (-1.25 - 0.407i)T + (59.0 + 42.9i)T^{2}
79 1+(11.2−8.15i)T+(24.4−75.1i)T2 1 + (11.2 - 8.15i)T + (24.4 - 75.1i)T^{2}
83 1+(−6.25+8.61i)T+(−25.6−78.9i)T2 1 + (-6.25 + 8.61i)T + (-25.6 - 78.9i)T^{2}
89 1−12.1T+89T2 1 - 12.1T + 89T^{2}
97 1+(−2.54−3.50i)T+(−29.9+92.2i)T2 1 + (-2.54 - 3.50i)T + (-29.9 + 92.2i)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

−11.25104920699014024535628216758, −10.40405986410559247905616987515, −9.651096649309886949928050708379, −8.811581589301565221043529469843, −7.74288048987982410470951713297, −6.37246817162784364028072001615, −5.44684792965018424306070369826, −3.39015195240389787076391756768, −2.60968400459934197277580531358, −0.48841012474344612065538495176, 2.71242486204635727632737602712, 3.83918975035262543886545957142, 5.74514001317296683980044063716, 6.60167059692584976616638858581, 7.44577066797481361968739535936, 8.629171223179739624400898301589, 9.367438909596232677706429556098, 9.928968616179725986163965310340, 11.61269928707709450967740524061, 12.41321062034327102830982866588

Graph of the ZZ-function along the critical line