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Properties

Label 2-2016-56.37-c1-0-6
Degree 22
Conductor 20162016
Sign 0.249−0.968i0.249 - 0.968i
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  + (−1.98 − 1.14i)5-s + (1.05 + 2.42i)7-s + (3.36 − 1.94i)11-s + 3.33i·13-s + (0.143 + 0.248i)17-s + (−2.41 − 1.39i)19-s + (−3.26 + 5.65i)23-s + (0.132 + 0.229i)25-s − 5.53i·29-s + (3.72 + 6.44i)31-s + (0.684 − 6.03i)35-s + (−5.15 − 2.97i)37-s + 3.51·41-s + 11.2i·43-s + (−0.0435 + 0.0753i)47-s + ⋯
L(s)  = 1  + (−0.888 − 0.513i)5-s + (0.399 + 0.916i)7-s + (1.01 − 0.585i)11-s + 0.924i·13-s + (0.0348 + 0.0603i)17-s + (−0.553 − 0.319i)19-s + (−0.681 + 1.17i)23-s + (0.0265 + 0.0459i)25-s − 1.02i·29-s + (0.668 + 1.15i)31-s + (0.115 − 1.01i)35-s + (−0.847 − 0.489i)37-s + 0.549·41-s + 1.71i·43-s + (−0.00634 + 0.0109i)47-s + ⋯

Functional equation

Λ(s)=(2016s/2 ΓC(s) L(s)=((0.249−0.968i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.249 - 0.968i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(2016s/2 ΓC(s+1/2) L(s)=((0.249−0.968i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.249 - 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 20162016    =    25⋅32⋅72^{5} \cdot 3^{2} \cdot 7
Sign: 0.249−0.968i0.249 - 0.968i
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), 0.249−0.968i)(2,\ 2016,\ (\ :1/2),\ 0.249 - 0.968i)

Particular Values

L(1)L(1) ≈\approx 1.2658859291.265885929
L(12)L(\frac12) ≈\approx 1.2658859291.265885929
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+(−1.05−2.42i)T 1 + (-1.05 - 2.42i)T
good5 1+(1.98+1.14i)T+(2.5+4.33i)T2 1 + (1.98 + 1.14i)T + (2.5 + 4.33i)T^{2}
11 1+(−3.36+1.94i)T+(5.5−9.52i)T2 1 + (-3.36 + 1.94i)T + (5.5 - 9.52i)T^{2}
13 1−3.33iT−13T2 1 - 3.33iT - 13T^{2}
17 1+(−0.143−0.248i)T+(−8.5+14.7i)T2 1 + (-0.143 - 0.248i)T + (-8.5 + 14.7i)T^{2}
19 1+(2.41+1.39i)T+(9.5+16.4i)T2 1 + (2.41 + 1.39i)T + (9.5 + 16.4i)T^{2}
23 1+(3.26−5.65i)T+(−11.5−19.9i)T2 1 + (3.26 - 5.65i)T + (-11.5 - 19.9i)T^{2}
29 1+5.53iT−29T2 1 + 5.53iT - 29T^{2}
31 1+(−3.72−6.44i)T+(−15.5+26.8i)T2 1 + (-3.72 - 6.44i)T + (-15.5 + 26.8i)T^{2}
37 1+(5.15+2.97i)T+(18.5+32.0i)T2 1 + (5.15 + 2.97i)T + (18.5 + 32.0i)T^{2}
41 1−3.51T+41T2 1 - 3.51T + 41T^{2}
43 1−11.2iT−43T2 1 - 11.2iT - 43T^{2}
47 1+(0.0435−0.0753i)T+(−23.5−40.7i)T2 1 + (0.0435 - 0.0753i)T + (-23.5 - 40.7i)T^{2}
53 1+(−6.11+3.52i)T+(26.5−45.8i)T2 1 + (-6.11 + 3.52i)T + (26.5 - 45.8i)T^{2}
59 1+(3.76−2.17i)T+(29.5−51.0i)T2 1 + (3.76 - 2.17i)T + (29.5 - 51.0i)T^{2}
61 1+(−6.20−3.58i)T+(30.5+52.8i)T2 1 + (-6.20 - 3.58i)T + (30.5 + 52.8i)T^{2}
67 1+(−11.2+6.51i)T+(33.5−58.0i)T2 1 + (-11.2 + 6.51i)T + (33.5 - 58.0i)T^{2}
71 1+6.18T+71T2 1 + 6.18T + 71T^{2}
73 1+(−6.93−12.0i)T+(−36.5+63.2i)T2 1 + (-6.93 - 12.0i)T + (-36.5 + 63.2i)T^{2}
79 1+(4.49−7.79i)T+(−39.5−68.4i)T2 1 + (4.49 - 7.79i)T + (-39.5 - 68.4i)T^{2}
83 1−17.6iT−83T2 1 - 17.6iT - 83T^{2}
89 1+(8.59−14.8i)T+(−44.5−77.0i)T2 1 + (8.59 - 14.8i)T + (-44.5 - 77.0i)T^{2}
97 1−6.46T+97T2 1 - 6.46T + 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

−9.180942602151750218014161574690, −8.455417427413407321195853142830, −8.047974729162207124406687574592, −6.91666141276085887136827553908, −6.17320089707176136304231873960, −5.26804573646279007524616449621, −4.30431167553405272204026809698, −3.72544722128777305718907154579, −2.39090754227849804580591932239, −1.21628276344347543894696023923, 0.50644242609538686433362103519, 1.92910518436731655474874149213, 3.30789156595552113746421595776, 4.02816267087850774593030575579, 4.67339051501010636505537032913, 5.91152491744437350105304157805, 6.86608328233631409986248997240, 7.36003515757297949690245172348, 8.135654200346407058011343257447, 8.803955549106158534372107711257

Graph of the ZZ-function along the critical line