↑

Properties

Label 2-275-11.4-c1-0-8
Degree 22
Conductor 275275
Sign 0.124−0.992i0.124 - 0.992i
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (1.85 + 1.34i)2-s + (−0.0131 + 0.0403i)3-s + (1.00 + 3.10i)4-s + (−0.0786 + 0.0571i)6-s + (0.352 + 1.08i)7-s + (−0.894 + 2.75i)8-s + (2.42 + 1.76i)9-s + (−1.53 − 2.94i)11-s − 0.138·12-s + (−2.27 − 1.65i)13-s + (−0.807 + 2.48i)14-s + (−0.0933 + 0.0677i)16-s + (−6.25 + 4.54i)17-s + (2.12 + 6.54i)18-s + (1.01 − 3.12i)19-s + ⋯
L(s)  = 1  + (1.31 + 0.953i)2-s + (−0.00756 + 0.0232i)3-s + (0.503 + 1.55i)4-s + (−0.0321 + 0.0233i)6-s + (0.133 + 0.409i)7-s + (−0.316 + 0.973i)8-s + (0.808 + 0.587i)9-s + (−0.461 − 0.887i)11-s − 0.0399·12-s + (−0.630 − 0.458i)13-s + (−0.215 + 0.664i)14-s + (−0.0233 + 0.0169i)16-s + (−1.51 + 1.10i)17-s + (0.500 + 1.54i)18-s + (0.232 − 0.716i)19-s + ⋯

Functional equation

Λ(s)=(275s/2 ΓC(s) L(s)=((0.124−0.992i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.124 - 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(275s/2 ΓC(s+1/2) L(s)=((0.124−0.992i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.124 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 275275    =    52⋅115^{2} \cdot 11
Sign: 0.124−0.992i0.124 - 0.992i
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), 0.124−0.992i)(2,\ 275,\ (\ :1/2),\ 0.124 - 0.992i)

Particular Values

L(1)L(1) ≈\approx 1.83943+1.62276i1.83943 + 1.62276i
L(12)L(\frac12) ≈\approx 1.83943+1.62276i1.83943 + 1.62276i
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+(1.53+2.94i)T 1 + (1.53 + 2.94i)T
good2 1+(−1.85−1.34i)T+(0.618+1.90i)T2 1 + (-1.85 - 1.34i)T + (0.618 + 1.90i)T^{2}
3 1+(0.0131−0.0403i)T+(−2.42−1.76i)T2 1 + (0.0131 - 0.0403i)T + (-2.42 - 1.76i)T^{2}
7 1+(−0.352−1.08i)T+(−5.66+4.11i)T2 1 + (-0.352 - 1.08i)T + (-5.66 + 4.11i)T^{2}
13 1+(2.27+1.65i)T+(4.01+12.3i)T2 1 + (2.27 + 1.65i)T + (4.01 + 12.3i)T^{2}
17 1+(6.25−4.54i)T+(5.25−16.1i)T2 1 + (6.25 - 4.54i)T + (5.25 - 16.1i)T^{2}
19 1+(−1.01+3.12i)T+(−15.3−11.1i)T2 1 + (-1.01 + 3.12i)T + (-15.3 - 11.1i)T^{2}
23 1−5.54T+23T2 1 - 5.54T + 23T^{2}
29 1+(2.15+6.62i)T+(−23.4+17.0i)T2 1 + (2.15 + 6.62i)T + (-23.4 + 17.0i)T^{2}
31 1+(−0.604−0.439i)T+(9.57+29.4i)T2 1 + (-0.604 - 0.439i)T + (9.57 + 29.4i)T^{2}
37 1+(1.48+4.57i)T+(−29.9+21.7i)T2 1 + (1.48 + 4.57i)T + (-29.9 + 21.7i)T^{2}
41 1+(−2.05+6.33i)T+(−33.1−24.0i)T2 1 + (-2.05 + 6.33i)T + (-33.1 - 24.0i)T^{2}
43 1+0.698T+43T2 1 + 0.698T + 43T^{2}
47 1+(2.67−8.22i)T+(−38.0−27.6i)T2 1 + (2.67 - 8.22i)T + (-38.0 - 27.6i)T^{2}
53 1+(7.08+5.15i)T+(16.3+50.4i)T2 1 + (7.08 + 5.15i)T + (16.3 + 50.4i)T^{2}
59 1+(−3.28−10.1i)T+(−47.7+34.6i)T2 1 + (-3.28 - 10.1i)T + (-47.7 + 34.6i)T^{2}
61 1+(7.48−5.44i)T+(18.8−58.0i)T2 1 + (7.48 - 5.44i)T + (18.8 - 58.0i)T^{2}
67 1+6.69T+67T2 1 + 6.69T + 67T^{2}
71 1+(6.03−4.38i)T+(21.9−67.5i)T2 1 + (6.03 - 4.38i)T + (21.9 - 67.5i)T^{2}
73 1+(−0.472−1.45i)T+(−59.0+42.9i)T2 1 + (-0.472 - 1.45i)T + (-59.0 + 42.9i)T^{2}
79 1+(−8.11−5.89i)T+(24.4+75.1i)T2 1 + (-8.11 - 5.89i)T + (24.4 + 75.1i)T^{2}
83 1+(3.96−2.87i)T+(25.6−78.9i)T2 1 + (3.96 - 2.87i)T + (25.6 - 78.9i)T^{2}
89 1+9.00T+89T2 1 + 9.00T + 89T^{2}
97 1+(−4.79−3.48i)T+(29.9+92.2i)T2 1 + (-4.79 - 3.48i)T + (29.9 + 92.2i)T^{2}
show more
show less
   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

−12.61541440065188737753662094844, −11.33969635246998792048045579129, −10.44630335755928040164508043097, −8.963985905820228941523849062952, −7.86888261191454271084334056052, −7.01236024335791306061062200200, −5.94052540430765995083695298701, −5.03585622215402594020540744992, −4.11736586961360434702180877804, −2.61321796592868141768839537120, 1.75101054373141819522033451894, 3.12472383524083026532441364917, 4.47812846930301418427266269486, 4.93957345454736225745727415280, 6.59134126644160080354931313787, 7.41717436174178072956722585776, 9.200678908570544980359053359496, 10.09151072638226298803465893164, 10.99695856663029243579408346275, 11.85745037998371182613858865707

Graph of the ZZ-function along the critical line