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Properties

Label 2-1456-7.4-c1-0-34
Degree 22
Conductor 14561456
Sign 0.605+0.795i0.605 + 0.795i
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  + (1.11 − 1.93i)3-s + (1.11 + 1.93i)5-s + (2 − 1.73i)7-s + (−1 − 1.73i)9-s + (−1.5 + 2.59i)11-s − 13-s + 5.00·15-s + (3.73 − 6.47i)17-s + (1.5 + 2.59i)19-s + (−1.11 − 5.80i)21-s + (−1.88 − 3.25i)23-s + 2.23·27-s − 4.47·29-s + (2.5 − 4.33i)31-s + (3.35 + 5.80i)33-s + ⋯
L(s)  = 1  + (0.645 − 1.11i)3-s + (0.499 + 0.866i)5-s + (0.755 − 0.654i)7-s + (−0.333 − 0.577i)9-s + (−0.452 + 0.783i)11-s − 0.277·13-s + 1.29·15-s + (0.906 − 1.56i)17-s + (0.344 + 0.596i)19-s + (−0.243 − 1.26i)21-s + (−0.392 − 0.679i)23-s + 0.430·27-s − 0.830·29-s + (0.449 − 0.777i)31-s + (0.583 + 1.01i)33-s + ⋯

Functional equation

Λ(s)=(1456s/2 ΓC(s) L(s)=((0.605+0.795i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1456s/2 ΓC(s+1/2) L(s)=((0.605+0.795i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 14561456    =    24⋅7⋅132^{4} \cdot 7 \cdot 13
Sign: 0.605+0.795i0.605 + 0.795i
Analytic conductor: 11.626211.6262
Root analytic conductor: 3.409723.40972
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1456(417,⋅)\chi_{1456} (417, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1456, ( :1/2), 0.605+0.795i)(2,\ 1456,\ (\ :1/2),\ 0.605 + 0.795i)

Particular Values

L(1)L(1) ≈\approx 2.5389987432.538998743
L(12)L(\frac12) ≈\approx 2.5389987432.538998743
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+(−2+1.73i)T 1 + (-2 + 1.73i)T
13 1+T 1 + T
good3 1+(−1.11+1.93i)T+(−1.5−2.59i)T2 1 + (-1.11 + 1.93i)T + (-1.5 - 2.59i)T^{2}
5 1+(−1.11−1.93i)T+(−2.5+4.33i)T2 1 + (-1.11 - 1.93i)T + (-2.5 + 4.33i)T^{2}
11 1+(1.5−2.59i)T+(−5.5−9.52i)T2 1 + (1.5 - 2.59i)T + (-5.5 - 9.52i)T^{2}
17 1+(−3.73+6.47i)T+(−8.5−14.7i)T2 1 + (-3.73 + 6.47i)T + (-8.5 - 14.7i)T^{2}
19 1+(−1.5−2.59i)T+(−9.5+16.4i)T2 1 + (-1.5 - 2.59i)T + (-9.5 + 16.4i)T^{2}
23 1+(1.88+3.25i)T+(−11.5+19.9i)T2 1 + (1.88 + 3.25i)T + (-11.5 + 19.9i)T^{2}
29 1+4.47T+29T2 1 + 4.47T + 29T^{2}
31 1+(−2.5+4.33i)T+(−15.5−26.8i)T2 1 + (-2.5 + 4.33i)T + (-15.5 - 26.8i)T^{2}
37 1+(−4.35−7.54i)T+(−18.5+32.0i)T2 1 + (-4.35 - 7.54i)T + (-18.5 + 32.0i)T^{2}
41 1−4.47T+41T2 1 - 4.47T + 41T^{2}
43 1−8T+43T2 1 - 8T + 43T^{2}
47 1+(−0.736−1.27i)T+(−23.5+40.7i)T2 1 + (-0.736 - 1.27i)T + (-23.5 + 40.7i)T^{2}
53 1+(0.736−1.27i)T+(−26.5−45.8i)T2 1 + (0.736 - 1.27i)T + (-26.5 - 45.8i)T^{2}
59 1+(−3.73+6.47i)T+(−29.5−51.0i)T2 1 + (-3.73 + 6.47i)T + (-29.5 - 51.0i)T^{2}
61 1+(1.5+2.59i)T+(−30.5+52.8i)T2 1 + (1.5 + 2.59i)T + (-30.5 + 52.8i)T^{2}
67 1+(1.5−2.59i)T+(−33.5−58.0i)T2 1 + (1.5 - 2.59i)T + (-33.5 - 58.0i)T^{2}
71 1+8.94T+71T2 1 + 8.94T + 71T^{2}
73 1+(5.35−9.27i)T+(−36.5−63.2i)T2 1 + (5.35 - 9.27i)T + (-36.5 - 63.2i)T^{2}
79 1+(−5.35−9.27i)T+(−39.5+68.4i)T2 1 + (-5.35 - 9.27i)T + (-39.5 + 68.4i)T^{2}
83 1+83T2 1 + 83T^{2}
89 1+(−1.11−1.93i)T+(−44.5+77.0i)T2 1 + (-1.11 - 1.93i)T + (-44.5 + 77.0i)T^{2}
97 1+17.4T+97T2 1 + 17.4T + 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

−9.564222070422563519301169369152, −8.241519480061036594486697281862, −7.59278680409530293448056594533, −7.26892796926927244876551351508, −6.40050956864377328904812130976, −5.30308062608915446196897066752, −4.28260959373508114776295645087, −2.83814372702961403383215518368, −2.29478276037662563121514322592, −1.10725356519076593075455531569, 1.38130982358383683771444019109, 2.64838452845639695590210996746, 3.69926878954685653487473631342, 4.57082295022008393980249525195, 5.48383092411319898480191329049, 5.87209179820867116533439991561, 7.59397953468103056663730610355, 8.264332197645826148036507641986, 9.064856202699270087965394092742, 9.311457896694164501359503092680

Graph of the ZZ-function along the critical line