↑

Properties

Label 2-234-117.41-c1-0-10
Degree 22
Conductor 234234
Sign 0.878+0.476i0.878 + 0.476i
Analytic cond. 1.868491.86849
Root an. cond. 1.366931.36693
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  + (−0.965 + 0.258i)2-s + (1.60 − 0.647i)3-s + (0.866 − 0.499i)4-s + (2.84 − 0.763i)5-s + (−1.38 + 1.04i)6-s + (−1.37 − 1.37i)7-s + (−0.707 + 0.707i)8-s + (2.16 − 2.07i)9-s + (−2.55 + 1.47i)10-s + (0.207 + 0.774i)11-s + (1.06 − 1.36i)12-s + (−3.58 − 0.369i)13-s + (1.68 + 0.975i)14-s + (4.08 − 3.06i)15-s + (0.500 − 0.866i)16-s + (−4.02 + 6.96i)17-s + ⋯
L(s)  = 1  + (−0.683 + 0.183i)2-s + (0.927 − 0.373i)3-s + (0.433 − 0.249i)4-s + (1.27 − 0.341i)5-s + (−0.565 + 0.424i)6-s + (−0.521 − 0.521i)7-s + (−0.249 + 0.249i)8-s + (0.720 − 0.693i)9-s + (−0.807 + 0.466i)10-s + (0.0625 + 0.233i)11-s + (0.308 − 0.393i)12-s + (−0.994 − 0.102i)13-s + (0.451 + 0.260i)14-s + (1.05 − 0.792i)15-s + (0.125 − 0.216i)16-s + (−0.975 + 1.68i)17-s + ⋯

Functional equation

Λ(s)=(234s/2 ΓC(s) L(s)=((0.878+0.476i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.878 + 0.476i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(234s/2 ΓC(s+1/2) L(s)=((0.878+0.476i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.878 + 0.476i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 234234    =    2⋅32⋅132 \cdot 3^{2} \cdot 13
Sign: 0.878+0.476i0.878 + 0.476i
Analytic conductor: 1.868491.86849
Root analytic conductor: 1.366931.36693
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ234(41,⋅)\chi_{234} (41, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 234, ( :1/2), 0.878+0.476i)(2,\ 234,\ (\ :1/2),\ 0.878 + 0.476i)

Particular Values

L(1)L(1) ≈\approx 1.31821−0.334631i1.31821 - 0.334631i
L(12)L(\frac12) ≈\approx 1.31821−0.334631i1.31821 - 0.334631i
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+(0.965−0.258i)T 1 + (0.965 - 0.258i)T
3 1+(−1.60+0.647i)T 1 + (-1.60 + 0.647i)T
13 1+(3.58+0.369i)T 1 + (3.58 + 0.369i)T
good5 1+(−2.84+0.763i)T+(4.33−2.5i)T2 1 + (-2.84 + 0.763i)T + (4.33 - 2.5i)T^{2}
7 1+(1.37+1.37i)T+7iT2 1 + (1.37 + 1.37i)T + 7iT^{2}
11 1+(−0.207−0.774i)T+(−9.52+5.5i)T2 1 + (-0.207 - 0.774i)T + (-9.52 + 5.5i)T^{2}
17 1+(4.02−6.96i)T+(−8.5−14.7i)T2 1 + (4.02 - 6.96i)T + (-8.5 - 14.7i)T^{2}
19 1+(1.18+4.41i)T+(−16.4+9.5i)T2 1 + (1.18 + 4.41i)T + (-16.4 + 9.5i)T^{2}
23 1−6.99T+23T2 1 - 6.99T + 23T^{2}
29 1+(−6.82−3.93i)T+(14.5+25.1i)T2 1 + (-6.82 - 3.93i)T + (14.5 + 25.1i)T^{2}
31 1+(−0.619−2.31i)T+(−26.8+15.5i)T2 1 + (-0.619 - 2.31i)T + (-26.8 + 15.5i)T^{2}
37 1+(2.09−7.82i)T+(−32.0−18.5i)T2 1 + (2.09 - 7.82i)T + (-32.0 - 18.5i)T^{2}
41 1+(3.66+3.66i)T+41iT2 1 + (3.66 + 3.66i)T + 41iT^{2}
43 1−2.33iT−43T2 1 - 2.33iT - 43T^{2}
47 1+(5.14+1.37i)T+(40.7+23.5i)T2 1 + (5.14 + 1.37i)T + (40.7 + 23.5i)T^{2}
53 1−8.27iT−53T2 1 - 8.27iT - 53T^{2}
59 1+(8.52+2.28i)T+(51.0+29.5i)T2 1 + (8.52 + 2.28i)T + (51.0 + 29.5i)T^{2}
61 1+4.93T+61T2 1 + 4.93T + 61T^{2}
67 1+(−0.549+0.549i)T−67iT2 1 + (-0.549 + 0.549i)T - 67iT^{2}
71 1+(2.57−0.689i)T+(61.4−35.5i)T2 1 + (2.57 - 0.689i)T + (61.4 - 35.5i)T^{2}
73 1+(−5.68−5.68i)T+73iT2 1 + (-5.68 - 5.68i)T + 73iT^{2}
79 1+(−0.554−0.960i)T+(−39.5+68.4i)T2 1 + (-0.554 - 0.960i)T + (-39.5 + 68.4i)T^{2}
83 1+(−0.614+2.29i)T+(−71.8−41.5i)T2 1 + (-0.614 + 2.29i)T + (-71.8 - 41.5i)T^{2}
89 1+(2.84+0.761i)T+(77.0+44.5i)T2 1 + (2.84 + 0.761i)T + (77.0 + 44.5i)T^{2}
97 1+(−2.59+2.59i)T−97iT2 1 + (-2.59 + 2.59i)T - 97iT^{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

−12.42943212484892516788898008334, −10.67191780647538184973093738397, −9.951418219643264636821172269098, −9.117095840521486538157897653617, −8.426672943273344445927162380366, −6.99516096156211890117155259340, −6.46911669981743307906616803411, −4.77930826791968926589559932029, −2.86963670248523741993018189944, −1.58144345385466811831595302942, 2.20951009244079053766876995144, 2.96753116998392828038895073325, 4.90646862467217129018284809716, 6.36661002261329958814286064036, 7.36926338038589454773030106427, 8.708153648266285168840934254018, 9.496283946850151939372933625213, 9.893572324423413702362766650158, 10.96076097048235364003894943500, 12.26949238350492626481044711186

Graph of the ZZ-function along the critical line