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Properties

Label 2-1456-13.4-c1-0-23
Degree 22
Conductor 14561456
Sign 0.991−0.129i0.991 - 0.129i
Analytic cond. 11.626211.6262
Root an. cond. 3.409723.40972
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  + (−1.41 + 2.44i)3-s + 0.518i·5-s + (0.866 − 0.5i)7-s + (−2.49 − 4.31i)9-s + (−1.40 − 0.812i)11-s + (1.42 − 3.31i)13-s + (−1.26 − 0.733i)15-s + (0.974 + 1.68i)17-s + (−2.15 + 1.24i)19-s + 2.82i·21-s + (4.57 − 7.91i)23-s + 4.73·25-s + 5.60·27-s + (2.61 − 4.52i)29-s − 5.79i·31-s + ⋯
L(s)  = 1  + (−0.815 + 1.41i)3-s + 0.232i·5-s + (0.327 − 0.188i)7-s + (−0.830 − 1.43i)9-s + (−0.424 − 0.244i)11-s + (0.395 − 0.918i)13-s + (−0.327 − 0.189i)15-s + (0.236 + 0.409i)17-s + (−0.494 + 0.285i)19-s + 0.616i·21-s + (0.952 − 1.65i)23-s + 0.946·25-s + 1.07·27-s + (0.485 − 0.841i)29-s − 1.04i·31-s + ⋯

Functional equation

Λ(s)=(1456s/2 ΓC(s) L(s)=((0.991−0.129i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.129i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1456s/2 ΓC(s+1/2) L(s)=((0.991−0.129i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 - 0.129i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 14561456    =    24⋅7⋅132^{4} \cdot 7 \cdot 13
Sign: 0.991−0.129i0.991 - 0.129i
Analytic conductor: 11.626211.6262
Root analytic conductor: 3.409723.40972
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1456(225,⋅)\chi_{1456} (225, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1456, ( :1/2), 0.991−0.129i)(2,\ 1456,\ (\ :1/2),\ 0.991 - 0.129i)

Particular Values

L(1)L(1) ≈\approx 1.1084284181.108428418
L(12)L(\frac12) ≈\approx 1.1084284181.108428418
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+(−0.866+0.5i)T 1 + (-0.866 + 0.5i)T
13 1+(−1.42+3.31i)T 1 + (-1.42 + 3.31i)T
good3 1+(1.41−2.44i)T+(−1.5−2.59i)T2 1 + (1.41 - 2.44i)T + (-1.5 - 2.59i)T^{2}
5 1−0.518iT−5T2 1 - 0.518iT - 5T^{2}
11 1+(1.40+0.812i)T+(5.5+9.52i)T2 1 + (1.40 + 0.812i)T + (5.5 + 9.52i)T^{2}
17 1+(−0.974−1.68i)T+(−8.5+14.7i)T2 1 + (-0.974 - 1.68i)T + (-8.5 + 14.7i)T^{2}
19 1+(2.15−1.24i)T+(9.5−16.4i)T2 1 + (2.15 - 1.24i)T + (9.5 - 16.4i)T^{2}
23 1+(−4.57+7.91i)T+(−11.5−19.9i)T2 1 + (-4.57 + 7.91i)T + (-11.5 - 19.9i)T^{2}
29 1+(−2.61+4.52i)T+(−14.5−25.1i)T2 1 + (-2.61 + 4.52i)T + (-14.5 - 25.1i)T^{2}
31 1+5.79iT−31T2 1 + 5.79iT - 31T^{2}
37 1+(8.85+5.11i)T+(18.5+32.0i)T2 1 + (8.85 + 5.11i)T + (18.5 + 32.0i)T^{2}
41 1+(−3.64−2.10i)T+(20.5+35.5i)T2 1 + (-3.64 - 2.10i)T + (20.5 + 35.5i)T^{2}
43 1+(−0.498−0.863i)T+(−21.5+37.2i)T2 1 + (-0.498 - 0.863i)T + (-21.5 + 37.2i)T^{2}
47 1+4.51iT−47T2 1 + 4.51iT - 47T^{2}
53 1+8.89T+53T2 1 + 8.89T + 53T^{2}
59 1+(−5.37+3.10i)T+(29.5−51.0i)T2 1 + (-5.37 + 3.10i)T + (29.5 - 51.0i)T^{2}
61 1+(−6.73−11.6i)T+(−30.5+52.8i)T2 1 + (-6.73 - 11.6i)T + (-30.5 + 52.8i)T^{2}
67 1+(−7.25−4.18i)T+(33.5+58.0i)T2 1 + (-7.25 - 4.18i)T + (33.5 + 58.0i)T^{2}
71 1+(−4.50+2.59i)T+(35.5−61.4i)T2 1 + (-4.50 + 2.59i)T + (35.5 - 61.4i)T^{2}
73 1−11.8iT−73T2 1 - 11.8iT - 73T^{2}
79 1+0.982T+79T2 1 + 0.982T + 79T^{2}
83 1−8.91iT−83T2 1 - 8.91iT - 83T^{2}
89 1+(10.4+6.00i)T+(44.5+77.0i)T2 1 + (10.4 + 6.00i)T + (44.5 + 77.0i)T^{2}
97 1+(−3.82+2.21i)T+(48.5−84.0i)T2 1 + (-3.82 + 2.21i)T + (48.5 - 84.0i)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

−9.868937880017665334029250255849, −8.726753687971771773697726389307, −8.195589950756270725176142087818, −6.93024114851158345461845494059, −5.99298022452267472988392814159, −5.32776171473176009808740325637, −4.50939273781209398515638121034, −3.73162594598680162853623328115, −2.65182442789504726519732955685, −0.59144387401092932543557465484, 1.10909008705558905456468872850, 1.92101656813415784775377751416, 3.25344095826931815841350051146, 4.89882774148071598088080197941, 5.29618827925500518521500842648, 6.48486873919834450845105602293, 6.94311845518354095331771373886, 7.69443384801146911673875131384, 8.603131985876215426754583693027, 9.307434418417840798766633412083

Graph of the ZZ-function along the critical line