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Properties

Label 2-42e2-28.23-c0-0-1
Degree 22
Conductor 17641764
Sign 0.605−0.795i0.605 - 0.795i
Analytic cond. 0.8803500.880350
Root an. cond. 0.9382700.938270
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  + (−0.866 − 0.5i)2-s + (0.499 + 0.866i)4-s − 0.999i·8-s + (−1.73 + i)11-s + (−0.5 + 0.866i)16-s + 1.99·22-s + (1.73 + i)23-s + (0.5 + 0.866i)25-s + (0.866 − 0.499i)32-s + (−1 + 1.73i)37-s + (−1.73 − 0.999i)44-s + (−0.999 − 1.73i)46-s − 0.999i·50-s − 0.999·64-s + 2i·71-s + ⋯
L(s)  = 1  + (−0.866 − 0.5i)2-s + (0.499 + 0.866i)4-s − 0.999i·8-s + (−1.73 + i)11-s + (−0.5 + 0.866i)16-s + 1.99·22-s + (1.73 + i)23-s + (0.5 + 0.866i)25-s + (0.866 − 0.499i)32-s + (−1 + 1.73i)37-s + (−1.73 − 0.999i)44-s + (−0.999 − 1.73i)46-s − 0.999i·50-s − 0.999·64-s + 2i·71-s + ⋯

Functional equation

Λ(s)=(1764s/2 ΓC(s) L(s)=((0.605−0.795i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(1764s/2 ΓC(s) L(s)=((0.605−0.795i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 17641764    =    22⋅32⋅722^{2} \cdot 3^{2} \cdot 7^{2}
Sign: 0.605−0.795i0.605 - 0.795i
Analytic conductor: 0.8803500.880350
Root analytic conductor: 0.9382700.938270
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ1764(667,⋅)\chi_{1764} (667, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1764, ( :0), 0.605−0.795i)(2,\ 1764,\ (\ :0),\ 0.605 - 0.795i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.58697055350.5869705535
L(12)L(\frac12) ≈\approx 0.58697055350.5869705535
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+(0.866+0.5i)T 1 + (0.866 + 0.5i)T
3 1 1
7 1 1
good5 1+(−0.5−0.866i)T2 1 + (-0.5 - 0.866i)T^{2}
11 1+(1.73−i)T+(0.5−0.866i)T2 1 + (1.73 - i)T + (0.5 - 0.866i)T^{2}
13 1+T2 1 + T^{2}
17 1+(−0.5+0.866i)T2 1 + (-0.5 + 0.866i)T^{2}
19 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
23 1+(−1.73−i)T+(0.5+0.866i)T2 1 + (-1.73 - i)T + (0.5 + 0.866i)T^{2}
29 1+T2 1 + T^{2}
31 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
37 1+(1−1.73i)T+(−0.5−0.866i)T2 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2}
41 1+T2 1 + T^{2}
43 1−T2 1 - T^{2}
47 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
53 1+(−0.5+0.866i)T2 1 + (-0.5 + 0.866i)T^{2}
59 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
61 1+(−0.5−0.866i)T2 1 + (-0.5 - 0.866i)T^{2}
67 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
71 1−2iT−T2 1 - 2iT - T^{2}
73 1+(−0.5+0.866i)T2 1 + (-0.5 + 0.866i)T^{2}
79 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
83 1−T2 1 - T^{2}
89 1+(−0.5−0.866i)T2 1 + (-0.5 - 0.866i)T^{2}
97 1+T2 1 + 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

−9.706769129691241505604179192285, −8.877359005296056440187324528902, −8.105007153093755029020923519337, −7.32152872094821419581859032643, −6.87157246585498388548340881002, −5.42616194856615565403755394025, −4.71714491947273489038642057406, −3.34078547643171601110707041047, −2.61889142530479702165042925502, −1.44193440244662016617745613553, 0.60097015769081178863827817118, 2.29251402830539862297653218458, 3.13222330033251035775996215260, 4.78997946875933757265319941348, 5.42966471062023955239344779615, 6.27756029136800055621069338263, 7.15292978823360098662071039641, 7.87451798494159881769975370341, 8.613029783060397991315509707163, 9.100777793763634865762488981956

Graph of the ZZ-function along the critical line