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Properties

Label 2-2016-56.53-c1-0-18
Degree 22
Conductor 20162016
Sign 0.944+0.329i0.944 + 0.329i
Analytic cond. 16.097816.0978
Root an. cond. 4.012214.01221
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  + (−1.56 + 0.902i)5-s + (−2.63 − 0.217i)7-s + (−4.48 − 2.58i)11-s + 0.840i·13-s + (−2.45 + 4.25i)17-s + (4.87 − 2.81i)19-s + (3.05 + 5.28i)23-s + (−0.872 + 1.51i)25-s − 0.439i·29-s + (3.66 − 6.35i)31-s + (4.31 − 2.03i)35-s + (4.56 − 2.63i)37-s + 6.23·41-s − 7.34i·43-s + (−2.83 − 4.91i)47-s + ⋯
L(s)  = 1  + (−0.698 + 0.403i)5-s + (−0.996 − 0.0820i)7-s + (−1.35 − 0.779i)11-s + 0.232i·13-s + (−0.595 + 1.03i)17-s + (1.11 − 0.646i)19-s + (0.636 + 1.10i)23-s + (−0.174 + 0.302i)25-s − 0.0816i·29-s + (0.658 − 1.14i)31-s + (0.729 − 0.344i)35-s + (0.750 − 0.433i)37-s + 0.974·41-s − 1.12i·43-s + (−0.413 − 0.716i)47-s + ⋯

Functional equation

Λ(s)=(2016s/2 ΓC(s) L(s)=((0.944+0.329i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.944 + 0.329i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(2016s/2 ΓC(s+1/2) L(s)=((0.944+0.329i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.944 + 0.329i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 20162016    =    25⋅32⋅72^{5} \cdot 3^{2} \cdot 7
Sign: 0.944+0.329i0.944 + 0.329i
Analytic conductor: 16.097816.0978
Root analytic conductor: 4.012214.01221
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ2016(1873,⋅)\chi_{2016} (1873, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2016, ( :1/2), 0.944+0.329i)(2,\ 2016,\ (\ :1/2),\ 0.944 + 0.329i)

Particular Values

L(1)L(1) ≈\approx 0.96283553670.9628355367
L(12)L(\frac12) ≈\approx 0.96283553670.9628355367
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
3 1 1
7 1+(2.63+0.217i)T 1 + (2.63 + 0.217i)T
good5 1+(1.56−0.902i)T+(2.5−4.33i)T2 1 + (1.56 - 0.902i)T + (2.5 - 4.33i)T^{2}
11 1+(4.48+2.58i)T+(5.5+9.52i)T2 1 + (4.48 + 2.58i)T + (5.5 + 9.52i)T^{2}
13 1−0.840iT−13T2 1 - 0.840iT - 13T^{2}
17 1+(2.45−4.25i)T+(−8.5−14.7i)T2 1 + (2.45 - 4.25i)T + (-8.5 - 14.7i)T^{2}
19 1+(−4.87+2.81i)T+(9.5−16.4i)T2 1 + (-4.87 + 2.81i)T + (9.5 - 16.4i)T^{2}
23 1+(−3.05−5.28i)T+(−11.5+19.9i)T2 1 + (-3.05 - 5.28i)T + (-11.5 + 19.9i)T^{2}
29 1+0.439iT−29T2 1 + 0.439iT - 29T^{2}
31 1+(−3.66+6.35i)T+(−15.5−26.8i)T2 1 + (-3.66 + 6.35i)T + (-15.5 - 26.8i)T^{2}
37 1+(−4.56+2.63i)T+(18.5−32.0i)T2 1 + (-4.56 + 2.63i)T + (18.5 - 32.0i)T^{2}
41 1−6.23T+41T2 1 - 6.23T + 41T^{2}
43 1+7.34iT−43T2 1 + 7.34iT - 43T^{2}
47 1+(2.83+4.91i)T+(−23.5+40.7i)T2 1 + (2.83 + 4.91i)T + (-23.5 + 40.7i)T^{2}
53 1+(−1.15−0.669i)T+(26.5+45.8i)T2 1 + (-1.15 - 0.669i)T + (26.5 + 45.8i)T^{2}
59 1+(7.31+4.22i)T+(29.5+51.0i)T2 1 + (7.31 + 4.22i)T + (29.5 + 51.0i)T^{2}
61 1+(−4.77+2.75i)T+(30.5−52.8i)T2 1 + (-4.77 + 2.75i)T + (30.5 - 52.8i)T^{2}
67 1+(0.647+0.373i)T+(33.5+58.0i)T2 1 + (0.647 + 0.373i)T + (33.5 + 58.0i)T^{2}
71 1−9.08T+71T2 1 - 9.08T + 71T^{2}
73 1+(−3.70+6.42i)T+(−36.5−63.2i)T2 1 + (-3.70 + 6.42i)T + (-36.5 - 63.2i)T^{2}
79 1+(−8.68−15.0i)T+(−39.5+68.4i)T2 1 + (-8.68 - 15.0i)T + (-39.5 + 68.4i)T^{2}
83 1+1.45iT−83T2 1 + 1.45iT - 83T^{2}
89 1+(−3.10−5.37i)T+(−44.5+77.0i)T2 1 + (-3.10 - 5.37i)T + (-44.5 + 77.0i)T^{2}
97 1+5.81T+97T2 1 + 5.81T + 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

−9.223044586864097454138993014638, −8.160120466880479903545850625793, −7.59510434321051968362106188195, −6.83278464270357073142677451253, −5.93570905232611173768601452731, −5.19911136692819664809890184172, −3.93397141523609961426310418601, −3.29416279509661885352814792642, −2.41257041107247033947309585816, −0.53798277873867917655961827067, 0.73137639419325880759795266377, 2.56066362397033143114884649117, 3.16358920622910205488037254709, 4.46815044917388350287662360637, 4.98454643009237927242169869094, 6.04444774397681339384685905792, 6.96262706379756645087635972680, 7.65735816340138737147085560154, 8.309443735393178186533612510317, 9.276655324239516876653548143709

Graph of the ZZ-function along the critical line