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Properties

Label 2-2016-504.499-c0-0-1
Degree 22
Conductor 20162016
Sign 0.971−0.235i0.971 - 0.235i
Analytic cond. 1.006111.00611
Root an. cond. 1.003051.00305
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  + 3-s + (0.866 − 0.5i)5-s + i·7-s + 9-s + (−0.5 + 0.866i)11-s + (−0.866 − 0.5i)13-s + (0.866 − 0.5i)15-s + (0.5 + 0.866i)17-s + (0.5 − 0.866i)19-s + i·21-s + (−0.866 + 0.5i)23-s + 27-s + (0.866 − 0.5i)29-s + (−0.5 + 0.866i)33-s + (0.5 + 0.866i)35-s + ⋯
L(s)  = 1  + 3-s + (0.866 − 0.5i)5-s + i·7-s + 9-s + (−0.5 + 0.866i)11-s + (−0.866 − 0.5i)13-s + (0.866 − 0.5i)15-s + (0.5 + 0.866i)17-s + (0.5 − 0.866i)19-s + i·21-s + (−0.866 + 0.5i)23-s + 27-s + (0.866 − 0.5i)29-s + (−0.5 + 0.866i)33-s + (0.5 + 0.866i)35-s + ⋯

Functional equation

Λ(s)=(2016s/2 ΓC(s) L(s)=((0.971−0.235i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.971 - 0.235i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(2016s/2 ΓC(s) L(s)=((0.971−0.235i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.971 - 0.235i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 20162016    =    25⋅32⋅72^{5} \cdot 3^{2} \cdot 7
Sign: 0.971−0.235i0.971 - 0.235i
Analytic conductor: 1.006111.00611
Root analytic conductor: 1.003051.00305
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ2016(751,⋅)\chi_{2016} (751, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2016, ( :0), 0.971−0.235i)(2,\ 2016,\ (\ :0),\ 0.971 - 0.235i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.8372938391.837293839
L(12)L(\frac12) ≈\approx 1.8372938391.837293839
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
3 1−T 1 - T
7 1−iT 1 - iT
good5 1+(−0.866+0.5i)T+(0.5−0.866i)T2 1 + (-0.866 + 0.5i)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+(0.866+0.5i)T+(0.5+0.866i)T2 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)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}
19 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+(0.866−0.5i)T+(0.5−0.866i)T2 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2}
29 1+(−0.866+0.5i)T+(0.5−0.866i)T2 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2}
31 1−T2 1 - T^{2}
37 1+(0.866+0.5i)T+(0.5+0.866i)T2 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2}
41 1+(−0.5+0.866i)T+(−0.5−0.866i)T2 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2}
43 1+(0.5+0.866i)T+(−0.5+0.866i)T2 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2}
47 1−T2 1 - T^{2}
53 1+(0.866−0.5i)T+(0.5−0.866i)T2 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2}
59 1+T2 1 + T^{2}
61 1−T2 1 - T^{2}
67 1+T2 1 + 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+2iT−T2 1 + 2iT - T^{2}
83 1+(0.5+0.866i)T+(−0.5+0.866i)T2 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2}
89 1+(0.5−0.866i)T+(−0.5−0.866i)T2 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2}
97 1+(0.5+0.866i)T+(−0.5+0.866i)T2 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)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

−9.288610713980954208986416134120, −8.750519803964511533521267146529, −7.85642503880076709538806310171, −7.27837812691924048384438245103, −6.09101298662472228554498440395, −5.30210141276762484865119996928, −4.62218694574782641427657427006, −3.33093884593254573198025812897, −2.36036768034826701041316288468, −1.76525679679166117442077359939, 1.41357876824931239496654712700, 2.60162087377778855620030738079, 3.25249313304339734490980712099, 4.30597100979190554303236632324, 5.23396914768327144724107978224, 6.36585551168249298336313573523, 7.01367441530388922669129005379, 7.84958737970590527664513956998, 8.379370142379055329173638219724, 9.574122687552991383256865577611

Graph of the ZZ-function along the critical line