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Properties

Label 2-275-11.4-c1-0-4
Degree 22
Conductor 275275
Sign −0.788−0.614i-0.788 - 0.614i
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  + (1.63 + 1.18i)2-s + (−0.809 + 2.49i)3-s + (0.647 + 1.99i)4-s + (−4.29 + 3.11i)6-s + (0.298 + 0.918i)7-s + (−0.0595 + 0.183i)8-s + (−3.12 − 2.27i)9-s + (−3.31 − 0.189i)11-s − 5.49·12-s + (3.66 + 2.65i)13-s + (−0.603 + 1.85i)14-s + (3.07 − 2.23i)16-s + (2.69 − 1.96i)17-s + (−2.41 − 7.44i)18-s + (1.01 − 3.11i)19-s + ⋯
L(s)  = 1  + (1.15 + 0.841i)2-s + (−0.467 + 1.43i)3-s + (0.323 + 0.996i)4-s + (−1.75 + 1.27i)6-s + (0.112 + 0.347i)7-s + (−0.0210 + 0.0648i)8-s + (−1.04 − 0.757i)9-s + (−0.998 − 0.0572i)11-s − 1.58·12-s + (1.01 + 0.737i)13-s + (−0.161 + 0.496i)14-s + (0.768 − 0.558i)16-s + (0.654 − 0.475i)17-s + (−0.570 − 1.75i)18-s + (0.232 − 0.715i)19-s + ⋯

Functional equation

Λ(s)=(275s/2 ΓC(s) L(s)=((−0.788−0.614i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.788 - 0.614i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(275s/2 ΓC(s+1/2) L(s)=((−0.788−0.614i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.788 - 0.614i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 275275    =    52⋅115^{2} \cdot 11
Sign: −0.788−0.614i-0.788 - 0.614i
Analytic conductor: 2.195882.19588
Root analytic conductor: 1.481851.48185
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ275(26,⋅)\chi_{275} (26, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 275, ( :1/2), −0.788−0.614i)(2,\ 275,\ (\ :1/2),\ -0.788 - 0.614i)

Particular Values

L(1)L(1) ≈\approx 0.632830+1.84204i0.632830 + 1.84204i
L(12)L(\frac12) ≈\approx 0.632830+1.84204i0.632830 + 1.84204i
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+(3.31+0.189i)T 1 + (3.31 + 0.189i)T
good2 1+(−1.63−1.18i)T+(0.618+1.90i)T2 1 + (-1.63 - 1.18i)T + (0.618 + 1.90i)T^{2}
3 1+(0.809−2.49i)T+(−2.42−1.76i)T2 1 + (0.809 - 2.49i)T + (-2.42 - 1.76i)T^{2}
7 1+(−0.298−0.918i)T+(−5.66+4.11i)T2 1 + (-0.298 - 0.918i)T + (-5.66 + 4.11i)T^{2}
13 1+(−3.66−2.65i)T+(4.01+12.3i)T2 1 + (-3.66 - 2.65i)T + (4.01 + 12.3i)T^{2}
17 1+(−2.69+1.96i)T+(5.25−16.1i)T2 1 + (-2.69 + 1.96i)T + (5.25 - 16.1i)T^{2}
19 1+(−1.01+3.11i)T+(−15.3−11.1i)T2 1 + (-1.01 + 3.11i)T + (-15.3 - 11.1i)T^{2}
23 1+3.36T+23T2 1 + 3.36T + 23T^{2}
29 1+(−1.51−4.67i)T+(−23.4+17.0i)T2 1 + (-1.51 - 4.67i)T + (-23.4 + 17.0i)T^{2}
31 1+(−0.338−0.245i)T+(9.57+29.4i)T2 1 + (-0.338 - 0.245i)T + (9.57 + 29.4i)T^{2}
37 1+(−1.95−6.02i)T+(−29.9+21.7i)T2 1 + (-1.95 - 6.02i)T + (-29.9 + 21.7i)T^{2}
41 1+(−1.78+5.50i)T+(−33.1−24.0i)T2 1 + (-1.78 + 5.50i)T + (-33.1 - 24.0i)T^{2}
43 1+2.26T+43T2 1 + 2.26T + 43T^{2}
47 1+(−1.33+4.11i)T+(−38.0−27.6i)T2 1 + (-1.33 + 4.11i)T + (-38.0 - 27.6i)T^{2}
53 1+(−2.15−1.56i)T+(16.3+50.4i)T2 1 + (-2.15 - 1.56i)T + (16.3 + 50.4i)T^{2}
59 1+(3.12+9.62i)T+(−47.7+34.6i)T2 1 + (3.12 + 9.62i)T + (-47.7 + 34.6i)T^{2}
61 1+(−1.99+1.45i)T+(18.8−58.0i)T2 1 + (-1.99 + 1.45i)T + (18.8 - 58.0i)T^{2}
67 1+9.60T+67T2 1 + 9.60T + 67T^{2}
71 1+(−4.41+3.20i)T+(21.9−67.5i)T2 1 + (-4.41 + 3.20i)T + (21.9 - 67.5i)T^{2}
73 1+(−0.443−1.36i)T+(−59.0+42.9i)T2 1 + (-0.443 - 1.36i)T + (-59.0 + 42.9i)T^{2}
79 1+(0.812+0.590i)T+(24.4+75.1i)T2 1 + (0.812 + 0.590i)T + (24.4 + 75.1i)T^{2}
83 1+(5.98−4.34i)T+(25.6−78.9i)T2 1 + (5.98 - 4.34i)T + (25.6 - 78.9i)T^{2}
89 1+12.1T+89T2 1 + 12.1T + 89T^{2}
97 1+(−2.44−1.77i)T+(29.9+92.2i)T2 1 + (-2.44 - 1.77i)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

−12.30875361669846668096535340780, −11.36233464780402505674761645503, −10.43527801724498256294824911229, −9.542094405218379472810446164616, −8.371999453372257984990945866770, −7.00037311118589519258632123271, −5.80518909109479622328173659078, −5.16801216453007885119829796792, −4.30332517529468013650245817438, −3.23316617100259924738788111020, 1.30367025921154396788616027944, 2.66871231753699609650735386782, 4.03033010007937574155635535886, 5.57560307468741288731835649773, 6.07586336647953913270150676694, 7.64951933533000909828904004373, 8.154016298530555672183671247874, 10.21475241176205291307325549511, 10.93961846685067281413221786610, 11.87408133961494299983365070371

Graph of the ZZ-function along the critical line