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Properties

Label 2-2016-56.37-c1-0-37
Degree 22
Conductor 20162016
Sign −0.550−0.834i-0.550 - 0.834i
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  + (−3.09 − 1.78i)5-s + (−0.993 − 2.45i)7-s + (−0.815 + 0.470i)11-s − 6.15i·13-s + (−1.89 − 3.27i)17-s + (−2.09 − 1.20i)19-s + (−1.49 + 2.58i)23-s + (3.90 + 6.75i)25-s − 2.68i·29-s + (5.35 + 9.27i)31-s + (−1.30 + 9.37i)35-s + (−1.47 − 0.853i)37-s + 4.55·41-s + 3.50i·43-s + (3.42 − 5.92i)47-s + ⋯
L(s)  = 1  + (−1.38 − 0.800i)5-s + (−0.375 − 0.926i)7-s + (−0.245 + 0.142i)11-s − 1.70i·13-s + (−0.458 − 0.794i)17-s + (−0.479 − 0.276i)19-s + (−0.311 + 0.539i)23-s + (0.780 + 1.35i)25-s − 0.497i·29-s + (0.962 + 1.66i)31-s + (−0.221 + 1.58i)35-s + (−0.243 − 0.140i)37-s + 0.712·41-s + 0.534i·43-s + (0.499 − 0.864i)47-s + ⋯

Functional equation

Λ(s)=(2016s/2 ΓC(s) L(s)=((−0.550−0.834i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.550 - 0.834i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(2016s/2 ΓC(s+1/2) L(s)=((−0.550−0.834i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.550 - 0.834i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 20162016    =    25⋅32⋅72^{5} \cdot 3^{2} \cdot 7
Sign: −0.550−0.834i-0.550 - 0.834i
Analytic conductor: 16.097816.0978
Root analytic conductor: 4.012214.01221
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ2016(1297,⋅)\chi_{2016} (1297, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2016, ( :1/2), −0.550−0.834i)(2,\ 2016,\ (\ :1/2),\ -0.550 - 0.834i)

Particular Values

L(1)L(1) ≈\approx 0.21899025410.2189902541
L(12)L(\frac12) ≈\approx 0.21899025410.2189902541
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+(0.993+2.45i)T 1 + (0.993 + 2.45i)T
good5 1+(3.09+1.78i)T+(2.5+4.33i)T2 1 + (3.09 + 1.78i)T + (2.5 + 4.33i)T^{2}
11 1+(0.815−0.470i)T+(5.5−9.52i)T2 1 + (0.815 - 0.470i)T + (5.5 - 9.52i)T^{2}
13 1+6.15iT−13T2 1 + 6.15iT - 13T^{2}
17 1+(1.89+3.27i)T+(−8.5+14.7i)T2 1 + (1.89 + 3.27i)T + (-8.5 + 14.7i)T^{2}
19 1+(2.09+1.20i)T+(9.5+16.4i)T2 1 + (2.09 + 1.20i)T + (9.5 + 16.4i)T^{2}
23 1+(1.49−2.58i)T+(−11.5−19.9i)T2 1 + (1.49 - 2.58i)T + (-11.5 - 19.9i)T^{2}
29 1+2.68iT−29T2 1 + 2.68iT - 29T^{2}
31 1+(−5.35−9.27i)T+(−15.5+26.8i)T2 1 + (-5.35 - 9.27i)T + (-15.5 + 26.8i)T^{2}
37 1+(1.47+0.853i)T+(18.5+32.0i)T2 1 + (1.47 + 0.853i)T + (18.5 + 32.0i)T^{2}
41 1−4.55T+41T2 1 - 4.55T + 41T^{2}
43 1−3.50iT−43T2 1 - 3.50iT - 43T^{2}
47 1+(−3.42+5.92i)T+(−23.5−40.7i)T2 1 + (-3.42 + 5.92i)T + (-23.5 - 40.7i)T^{2}
53 1+(6.57−3.79i)T+(26.5−45.8i)T2 1 + (6.57 - 3.79i)T + (26.5 - 45.8i)T^{2}
59 1+(−0.100+0.0580i)T+(29.5−51.0i)T2 1 + (-0.100 + 0.0580i)T + (29.5 - 51.0i)T^{2}
61 1+(7.06+4.07i)T+(30.5+52.8i)T2 1 + (7.06 + 4.07i)T + (30.5 + 52.8i)T^{2}
67 1+(−3.44+1.98i)T+(33.5−58.0i)T2 1 + (-3.44 + 1.98i)T + (33.5 - 58.0i)T^{2}
71 1+3.92T+71T2 1 + 3.92T + 71T^{2}
73 1+(3.11+5.39i)T+(−36.5+63.2i)T2 1 + (3.11 + 5.39i)T + (-36.5 + 63.2i)T^{2}
79 1+(2.73−4.73i)T+(−39.5−68.4i)T2 1 + (2.73 - 4.73i)T + (-39.5 - 68.4i)T^{2}
83 1−1.19iT−83T2 1 - 1.19iT - 83T^{2}
89 1+(0.910−1.57i)T+(−44.5−77.0i)T2 1 + (0.910 - 1.57i)T + (-44.5 - 77.0i)T^{2}
97 1−12.0T+97T2 1 - 12.0T + 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

−8.469360453646926121215598893786, −7.80363454787156933830160529772, −7.36439248776337786132452335751, −6.36228938578472802156740149830, −5.14787700090367292553681660381, −4.55673175891771799386234746683, −3.66628838584589434476343249611, −2.88124830988028118405531493210, −0.987345270150443266771459939789, −0.095540317539969997267137557944, 2.04145193610057444723399261768, 2.94919844914497573069756586049, 4.02037674503992543534092747350, 4.48229280632995034050460360217, 5.97788128749959610023222458700, 6.50579238197460407885962899498, 7.30450565903187825700262694589, 8.167131384333525186774040394773, 8.745335682643055097290101004844, 9.569082324457402548038006895224

Graph of the ZZ-function along the critical line