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Properties

Label 2-2156-7.2-c1-0-18
Degree 22
Conductor 21562156
Sign 0.991+0.126i0.991 + 0.126i
Analytic cond. 17.215717.2157
Root an. cond. 4.149184.14918
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  + (0.160 + 0.278i)3-s + (−1.94 + 3.37i)5-s + (1.44 − 2.50i)9-s + (−0.5 − 0.866i)11-s − 0.218·13-s − 1.25·15-s + (−2.43 − 4.21i)17-s + (0.321 − 0.557i)19-s + (3.26 − 5.66i)23-s + (−5.09 − 8.81i)25-s + 1.89·27-s + 10.4·29-s + (1.51 + 2.62i)31-s + (0.160 − 0.278i)33-s + (0.948 − 1.64i)37-s + ⋯
L(s)  = 1  + (0.0928 + 0.160i)3-s + (−0.871 + 1.50i)5-s + (0.482 − 0.836i)9-s + (−0.150 − 0.261i)11-s − 0.0605·13-s − 0.323·15-s + (−0.589 − 1.02i)17-s + (0.0737 − 0.127i)19-s + (0.681 − 1.18i)23-s + (−1.01 − 1.76i)25-s + 0.364·27-s + 1.93·29-s + (0.272 + 0.472i)31-s + (0.0279 − 0.0484i)33-s + (0.155 − 0.270i)37-s + ⋯

Functional equation

Λ(s)=(2156s/2 ΓC(s) L(s)=((0.991+0.126i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 2156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 + 0.126i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(2156s/2 ΓC(s+1/2) L(s)=((0.991+0.126i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2156 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 + 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 21562156    =    22⋅72⋅112^{2} \cdot 7^{2} \cdot 11
Sign: 0.991+0.126i0.991 + 0.126i
Analytic conductor: 17.215717.2157
Root analytic conductor: 4.149184.14918
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ2156(177,⋅)\chi_{2156} (177, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2156, ( :1/2), 0.991+0.126i)(2,\ 2156,\ (\ :1/2),\ 0.991 + 0.126i)

Particular Values

L(1)L(1) ≈\approx 1.4262574931.426257493
L(12)L(\frac12) ≈\approx 1.4262574931.426257493
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
7 1 1
11 1+(0.5+0.866i)T 1 + (0.5 + 0.866i)T
good3 1+(−0.160−0.278i)T+(−1.5+2.59i)T2 1 + (-0.160 - 0.278i)T + (-1.5 + 2.59i)T^{2}
5 1+(1.94−3.37i)T+(−2.5−4.33i)T2 1 + (1.94 - 3.37i)T + (-2.5 - 4.33i)T^{2}
13 1+0.218T+13T2 1 + 0.218T + 13T^{2}
17 1+(2.43+4.21i)T+(−8.5+14.7i)T2 1 + (2.43 + 4.21i)T + (-8.5 + 14.7i)T^{2}
19 1+(−0.321+0.557i)T+(−9.5−16.4i)T2 1 + (-0.321 + 0.557i)T + (-9.5 - 16.4i)T^{2}
23 1+(−3.26+5.66i)T+(−11.5−19.9i)T2 1 + (-3.26 + 5.66i)T + (-11.5 - 19.9i)T^{2}
29 1−10.4T+29T2 1 - 10.4T + 29T^{2}
31 1+(−1.51−2.62i)T+(−15.5+26.8i)T2 1 + (-1.51 - 2.62i)T + (-15.5 + 26.8i)T^{2}
37 1+(−0.948+1.64i)T+(−18.5−32.0i)T2 1 + (-0.948 + 1.64i)T + (-18.5 - 32.0i)T^{2}
41 1+4.86T+41T2 1 + 4.86T + 41T^{2}
43 1+1.35T+43T2 1 + 1.35T + 43T^{2}
47 1+(−2.78+4.82i)T+(−23.5−40.7i)T2 1 + (-2.78 + 4.82i)T + (-23.5 - 40.7i)T^{2}
53 1+(−3.89−6.74i)T+(−26.5+45.8i)T2 1 + (-3.89 - 6.74i)T + (-26.5 + 45.8i)T^{2}
59 1+(−3.83−6.64i)T+(−29.5+51.0i)T2 1 + (-3.83 - 6.64i)T + (-29.5 + 51.0i)T^{2}
61 1+(−4.00+6.93i)T+(−30.5−52.8i)T2 1 + (-4.00 + 6.93i)T + (-30.5 - 52.8i)T^{2}
67 1+(1.26+2.19i)T+(−33.5+58.0i)T2 1 + (1.26 + 2.19i)T + (-33.5 + 58.0i)T^{2}
71 1+14.3T+71T2 1 + 14.3T + 71T^{2}
73 1+(−4.43−7.67i)T+(−36.5+63.2i)T2 1 + (-4.43 - 7.67i)T + (-36.5 + 63.2i)T^{2}
79 1+(6.57−11.3i)T+(−39.5−68.4i)T2 1 + (6.57 - 11.3i)T + (-39.5 - 68.4i)T^{2}
83 1−9.72T+83T2 1 - 9.72T + 83T^{2}
89 1+(−5.30+9.18i)T+(−44.5−77.0i)T2 1 + (-5.30 + 9.18i)T + (-44.5 - 77.0i)T^{2}
97 1−0.746T+97T2 1 - 0.746T + 97T^{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

−8.954230986069185474751728195966, −8.331584983545290254194983829200, −7.22058892981886070720661296939, −6.88058111322526611873778353957, −6.24170336968499550473430869006, −4.84454343867032426761033347930, −4.08187080413008127069325662175, −3.11066521037368758677767850212, −2.60807007762075833575688387093, −0.63376028209176066330298216705, 0.990553545475000579962355740144, 1.97881037082722025692354397981, 3.42603938634556892874002736490, 4.51440583412259033568327453378, 4.78185682215433250577209714816, 5.79584642693067227029280620183, 6.95554026952207755992985090767, 7.72963587311613982553051174851, 8.341282070037325762990030861303, 8.820801911501504162237685163469

Graph of the ZZ-function along the critical line