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Properties

Label 2-39e2-13.2-c0-0-3
Degree 22
Conductor 15211521
Sign 0.999−0.0386i0.999 - 0.0386i
Analytic cond. 0.7590770.759077
Root an. cond. 0.8712500.871250
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  + (1.36 + 0.366i)2-s + (0.866 + 0.5i)4-s + (1 − i)5-s + (1.73 − i)10-s + (−0.366 + 1.36i)11-s + (−0.499 − 0.866i)16-s + (1.36 − 0.366i)20-s + (−1 + 1.73i)22-s − i·25-s + (−0.366 − 1.36i)32-s + (−1.36 − 0.366i)41-s + (1.73 + i)43-s + (−1 + 0.999i)44-s + (−1 − i)47-s + (−0.866 + 0.5i)49-s + (0.366 − 1.36i)50-s + ⋯
L(s)  = 1  + (1.36 + 0.366i)2-s + (0.866 + 0.5i)4-s + (1 − i)5-s + (1.73 − i)10-s + (−0.366 + 1.36i)11-s + (−0.499 − 0.866i)16-s + (1.36 − 0.366i)20-s + (−1 + 1.73i)22-s − i·25-s + (−0.366 − 1.36i)32-s + (−1.36 − 0.366i)41-s + (1.73 + i)43-s + (−1 + 0.999i)44-s + (−1 − i)47-s + (−0.866 + 0.5i)49-s + (0.366 − 1.36i)50-s + ⋯

Functional equation

Λ(s)=(1521s/2 ΓC(s) L(s)=((0.999−0.0386i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0386i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(1521s/2 ΓC(s) L(s)=((0.999−0.0386i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0386i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 15211521    =    32⋅1323^{2} \cdot 13^{2}
Sign: 0.999−0.0386i0.999 - 0.0386i
Analytic conductor: 0.7590770.759077
Root analytic conductor: 0.8712500.871250
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ1521(1432,⋅)\chi_{1521} (1432, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1521, ( :0), 0.999−0.0386i)(2,\ 1521,\ (\ :0),\ 0.999 - 0.0386i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 2.4271401412.427140141
L(12)L(\frac12) ≈\approx 2.4271401412.427140141
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)
bad3 1 1
13 1 1
good2 1+(−1.36−0.366i)T+(0.866+0.5i)T2 1 + (-1.36 - 0.366i)T + (0.866 + 0.5i)T^{2}
5 1+(−1+i)T−iT2 1 + (-1 + i)T - iT^{2}
7 1+(0.866−0.5i)T2 1 + (0.866 - 0.5i)T^{2}
11 1+(0.366−1.36i)T+(−0.866−0.5i)T2 1 + (0.366 - 1.36i)T + (-0.866 - 0.5i)T^{2}
17 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
19 1+(−0.866+0.5i)T2 1 + (-0.866 + 0.5i)T^{2}
23 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
29 1+(−0.5+0.866i)T2 1 + (-0.5 + 0.866i)T^{2}
31 1−iT2 1 - iT^{2}
37 1+(−0.866−0.5i)T2 1 + (-0.866 - 0.5i)T^{2}
41 1+(1.36+0.366i)T+(0.866+0.5i)T2 1 + (1.36 + 0.366i)T + (0.866 + 0.5i)T^{2}
43 1+(−1.73−i)T+(0.5+0.866i)T2 1 + (-1.73 - i)T + (0.5 + 0.866i)T^{2}
47 1+(1+i)T+iT2 1 + (1 + i)T + iT^{2}
53 1+T2 1 + T^{2}
59 1+(1.36−0.366i)T+(0.866−0.5i)T2 1 + (1.36 - 0.366i)T + (0.866 - 0.5i)T^{2}
61 1+(−0.5−0.866i)T2 1 + (-0.5 - 0.866i)T^{2}
67 1+(0.866+0.5i)T2 1 + (0.866 + 0.5i)T^{2}
71 1+(−0.366−1.36i)T+(−0.866+0.5i)T2 1 + (-0.366 - 1.36i)T + (-0.866 + 0.5i)T^{2}
73 1+iT2 1 + iT^{2}
79 1+T2 1 + T^{2}
83 1+(1−i)T−iT2 1 + (1 - i)T - iT^{2}
89 1+(−0.366+1.36i)T+(−0.866−0.5i)T2 1 + (-0.366 + 1.36i)T + (-0.866 - 0.5i)T^{2}
97 1+(−0.866+0.5i)T2 1 + (-0.866 + 0.5i)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

−9.627295815795258291399525624070, −9.010093944693941763344673284335, −7.87894053040409549726745463023, −6.95226163330634574248551785687, −6.16528671611276756694382475602, −5.31946055384630044631537323867, −4.84672972981776856958144039799, −4.07800100687861798798968593928, −2.75928812159246748507520268650, −1.66788522944546560811375725694, 1.93216772732673882481633639646, 2.94371747653891445625218477173, 3.41862342013857292150232990428, 4.66548507964020217502891770608, 5.61110077237492478153073272798, 6.10024487046170048702413973472, 6.79381963602542819335994620996, 8.009555252667046027492794295819, 8.937656671001005100145853633412, 9.880872982848770537550837972148

Graph of the ZZ-function along the critical line