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Properties

Label 2-2128-133.18-c0-0-4
Degree 22
Conductor 21282128
Sign −0.895+0.444i-0.895 + 0.444i
Analytic cond. 1.062011.06201
Root an. cond. 1.030531.03053
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1 − 1.73i)5-s + (0.5 − 0.866i)7-s + (−0.5 − 0.866i)9-s + (−0.5 + 0.866i)11-s + (0.5 − 0.866i)17-s + (0.5 + 0.866i)19-s + (−0.5 − 0.866i)23-s + (−1.49 + 2.59i)25-s − 1.99·35-s − 2·43-s + (−0.999 + 1.73i)45-s + (−0.5 − 0.866i)47-s + (−0.499 − 0.866i)49-s + 1.99·55-s + (0.5 + 0.866i)61-s + ⋯
L(s)  = 1  + (−1 − 1.73i)5-s + (0.5 − 0.866i)7-s + (−0.5 − 0.866i)9-s + (−0.5 + 0.866i)11-s + (0.5 − 0.866i)17-s + (0.5 + 0.866i)19-s + (−0.5 − 0.866i)23-s + (−1.49 + 2.59i)25-s − 1.99·35-s − 2·43-s + (−0.999 + 1.73i)45-s + (−0.5 − 0.866i)47-s + (−0.499 − 0.866i)49-s + 1.99·55-s + (0.5 + 0.866i)61-s + ⋯

Functional equation

Λ(s)=(2128s/2 ΓC(s) L(s)=((−0.895+0.444i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.895 + 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(2128s/2 ΓC(s) L(s)=((−0.895+0.444i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.895 + 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 21282128    =    24⋅7⋅192^{4} \cdot 7 \cdot 19
Sign: −0.895+0.444i-0.895 + 0.444i
Analytic conductor: 1.062011.06201
Root analytic conductor: 1.030531.03053
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ2128(417,⋅)\chi_{2128} (417, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2128, ( :0), −0.895+0.444i)(2,\ 2128,\ (\ :0),\ -0.895 + 0.444i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.72136941600.7213694160
L(12)L(\frac12) ≈\approx 0.72136941600.7213694160
L(1)L(1) 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+(−0.5+0.866i)T 1 + (-0.5 + 0.866i)T
19 1+(−0.5−0.866i)T 1 + (-0.5 - 0.866i)T
good3 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
5 1+(1+1.73i)T+(−0.5+0.866i)T2 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2}
11 1+(0.5−0.866i)T+(−0.5−0.866i)T2 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2}
13 1−T2 1 - T^{2}
17 1+(−0.5+0.866i)T+(−0.5−0.866i)T2 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2}
23 1+(0.5+0.866i)T+(−0.5+0.866i)T2 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2}
29 1−T2 1 - T^{2}
31 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
37 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
41 1−T2 1 - T^{2}
43 1+2T+T2 1 + 2T + T^{2}
47 1+(0.5+0.866i)T+(−0.5+0.866i)T2 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2}
53 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
59 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
61 1+(−0.5−0.866i)T+(−0.5+0.866i)T2 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2}
67 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
71 1−T2 1 - T^{2}
73 1+(−0.5+0.866i)T+(−0.5−0.866i)T2 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2}
79 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
83 1−T+T2 1 - T + T^{2}
89 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
97 1−T2 1 - 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

−8.813419084123669020957093842692, −8.117961030384810626376934711731, −7.66422529883835850432066947395, −6.77677885578494512395463355894, −5.44550877662521242518424498477, −4.88610224268032303691317605872, −4.13482737897399672125272523849, −3.37400144630158216670578194209, −1.62723535878004301495587793038, −0.50808988783476093579504957568, 2.12231506943909371368472235352, 2.99957676295888914589009525769, 3.56711007551147313958582428282, 4.88660129707229311740319485909, 5.73740428519226720156244440612, 6.47438792942762629308543098369, 7.46080611846657440068459729531, 8.065844195349755557467447390175, 8.453448668459997295752894324619, 9.746940635088989142203619801509

Graph of the ZZ-function along the critical line