↑

Properties

Label 2-88-11.5-c1-0-1
Degree 22
Conductor 8888
Sign 0.944−0.329i0.944 - 0.329i
Analytic cond. 0.7026830.702683
Root an. cond. 0.8382620.838262
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  + (1.30 + 0.951i)3-s + (0.190 − 0.587i)5-s + (−1.30 + 0.951i)7-s + (−0.118 − 0.363i)9-s + (1.23 + 3.07i)11-s + (−1.80 − 5.56i)13-s + (0.809 − 0.587i)15-s + (−0.572 + 1.76i)17-s + (−3.92 − 2.85i)19-s − 2.61·21-s − 4·23-s + (3.73 + 2.71i)25-s + (1.69 − 5.20i)27-s + (−5.92 + 4.30i)29-s + (0.336 + 1.03i)31-s + ⋯
L(s)  = 1  + (0.755 + 0.549i)3-s + (0.0854 − 0.262i)5-s + (−0.494 + 0.359i)7-s + (−0.0393 − 0.121i)9-s + (0.372 + 0.927i)11-s + (−0.501 − 1.54i)13-s + (0.208 − 0.151i)15-s + (−0.138 + 0.427i)17-s + (−0.900 − 0.654i)19-s − 0.571·21-s − 0.834·23-s + (0.747 + 0.542i)25-s + (0.325 − 1.00i)27-s + (−1.10 + 0.799i)29-s + (0.0605 + 0.186i)31-s + ⋯

Functional equation

Λ(s)=(88s/2 ΓC(s) L(s)=((0.944−0.329i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 88 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.944 - 0.329i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(88s/2 ΓC(s+1/2) L(s)=((0.944−0.329i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 88 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.944 - 0.329i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 8888    =    23⋅112^{3} \cdot 11
Sign: 0.944−0.329i0.944 - 0.329i
Analytic conductor: 0.7026830.702683
Root analytic conductor: 0.8382620.838262
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ88(49,⋅)\chi_{88} (49, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 88, ( :1/2), 0.944−0.329i)(2,\ 88,\ (\ :1/2),\ 0.944 - 0.329i)

Particular Values

L(1)L(1) ≈\approx 1.12108+0.190218i1.12108 + 0.190218i
L(12)L(\frac12) ≈\approx 1.12108+0.190218i1.12108 + 0.190218i
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)
bad2 1 1
11 1+(−1.23−3.07i)T 1 + (-1.23 - 3.07i)T
good3 1+(−1.30−0.951i)T+(0.927+2.85i)T2 1 + (-1.30 - 0.951i)T + (0.927 + 2.85i)T^{2}
5 1+(−0.190+0.587i)T+(−4.04−2.93i)T2 1 + (-0.190 + 0.587i)T + (-4.04 - 2.93i)T^{2}
7 1+(1.30−0.951i)T+(2.16−6.65i)T2 1 + (1.30 - 0.951i)T + (2.16 - 6.65i)T^{2}
13 1+(1.80+5.56i)T+(−10.5+7.64i)T2 1 + (1.80 + 5.56i)T + (-10.5 + 7.64i)T^{2}
17 1+(0.572−1.76i)T+(−13.7−9.99i)T2 1 + (0.572 - 1.76i)T + (-13.7 - 9.99i)T^{2}
19 1+(3.92+2.85i)T+(5.87+18.0i)T2 1 + (3.92 + 2.85i)T + (5.87 + 18.0i)T^{2}
23 1+4T+23T2 1 + 4T + 23T^{2}
29 1+(5.92−4.30i)T+(8.96−27.5i)T2 1 + (5.92 - 4.30i)T + (8.96 - 27.5i)T^{2}
31 1+(−0.336−1.03i)T+(−25.0+18.2i)T2 1 + (-0.336 - 1.03i)T + (-25.0 + 18.2i)T^{2}
37 1+(−7.78+5.65i)T+(11.4−35.1i)T2 1 + (-7.78 + 5.65i)T + (11.4 - 35.1i)T^{2}
41 1+(−7.78−5.65i)T+(12.6+38.9i)T2 1 + (-7.78 - 5.65i)T + (12.6 + 38.9i)T^{2}
43 1−1.52T+43T2 1 - 1.52T + 43T^{2}
47 1+(−8.54−6.20i)T+(14.5+44.6i)T2 1 + (-8.54 - 6.20i)T + (14.5 + 44.6i)T^{2}
53 1+(−0.190−0.587i)T+(−42.8+31.1i)T2 1 + (-0.190 - 0.587i)T + (-42.8 + 31.1i)T^{2}
59 1+(−1.92+1.40i)T+(18.2−56.1i)T2 1 + (-1.92 + 1.40i)T + (18.2 - 56.1i)T^{2}
61 1+(0.572−1.76i)T+(−49.3−35.8i)T2 1 + (0.572 - 1.76i)T + (-49.3 - 35.8i)T^{2}
67 1+14.4T+67T2 1 + 14.4T + 67T^{2}
71 1+(1.57−4.84i)T+(−57.4−41.7i)T2 1 + (1.57 - 4.84i)T + (-57.4 - 41.7i)T^{2}
73 1+(−2.54+1.84i)T+(22.5−69.4i)T2 1 + (-2.54 + 1.84i)T + (22.5 - 69.4i)T^{2}
79 1+(1.19+3.66i)T+(−63.9+46.4i)T2 1 + (1.19 + 3.66i)T + (-63.9 + 46.4i)T^{2}
83 1+(−2.89+8.92i)T+(−67.1−48.7i)T2 1 + (-2.89 + 8.92i)T + (-67.1 - 48.7i)T^{2}
89 1+4.47T+89T2 1 + 4.47T + 89T^{2}
97 1+(3.80+11.7i)T+(−78.4+57.0i)T2 1 + (3.80 + 11.7i)T + (-78.4 + 57.0i)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

−14.63728589141201142017890207983, −12.97063683353763014490457277602, −12.44868248569036383095529520014, −10.75783256480899482792160194696, −9.625390709378512383913604151011, −8.912621926737496808017444110918, −7.57722725373452583130631703521, −5.95159987271859050498514638788, −4.31896050633542380036399026447, −2.78718946515323995664647819920, 2.34078975853603948679166977390, 4.05643551663732213738238481037, 6.18444278864990889391846223786, 7.28270838749890953004482416683, 8.497903143780048054960115534978, 9.549394276781670604683352469790, 10.90525576995625567920765414624, 12.06882060277819161099230416187, 13.36291278171600563506987517234, 14.00909561356033508879730161101

Graph of the ZZ-function along the critical line