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Properties

Label 2-2100-105.68-c0-0-0
Degree 22
Conductor 21002100
Sign 0.0560−0.998i0.0560 - 0.998i
Analytic cond. 1.048031.04803
Root an. cond. 1.023731.02373
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  + (−0.965 − 0.258i)3-s + (0.707 + 0.707i)7-s + (0.866 + 0.499i)9-s + (−1.41 + 1.41i)13-s + (−0.500 − 0.866i)21-s + (−0.707 − 0.707i)27-s + (−1.5 + 0.866i)31-s + (0.448 + 1.67i)37-s + (1.73 − 1.00i)39-s + (−1.22 − 1.22i)43-s + 1.00i·49-s + (1.5 + 0.866i)61-s + (0.258 + 0.965i)63-s + (0.965 + 0.258i)73-s + (0.866 + 0.5i)79-s + ⋯
L(s)  = 1  + (−0.965 − 0.258i)3-s + (0.707 + 0.707i)7-s + (0.866 + 0.499i)9-s + (−1.41 + 1.41i)13-s + (−0.500 − 0.866i)21-s + (−0.707 − 0.707i)27-s + (−1.5 + 0.866i)31-s + (0.448 + 1.67i)37-s + (1.73 − 1.00i)39-s + (−1.22 − 1.22i)43-s + 1.00i·49-s + (1.5 + 0.866i)61-s + (0.258 + 0.965i)63-s + (0.965 + 0.258i)73-s + (0.866 + 0.5i)79-s + ⋯

Functional equation

Λ(s)=(2100s/2 ΓC(s) L(s)=((0.0560−0.998i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0560 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(2100s/2 ΓC(s) L(s)=((0.0560−0.998i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0560 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 21002100    =    22⋅3⋅52⋅72^{2} \cdot 3 \cdot 5^{2} \cdot 7
Sign: 0.0560−0.998i0.0560 - 0.998i
Analytic conductor: 1.048031.04803
Root analytic conductor: 1.023731.02373
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ2100(593,⋅)\chi_{2100} (593, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2100, ( :0), 0.0560−0.998i)(2,\ 2100,\ (\ :0),\ 0.0560 - 0.998i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.69210733320.6921073332
L(12)L(\frac12) ≈\approx 0.69210733320.6921073332
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+(0.965+0.258i)T 1 + (0.965 + 0.258i)T
5 1 1
7 1+(−0.707−0.707i)T 1 + (-0.707 - 0.707i)T
good11 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
13 1+(1.41−1.41i)T−iT2 1 + (1.41 - 1.41i)T - iT^{2}
17 1+(0.866+0.5i)T2 1 + (0.866 + 0.5i)T^{2}
19 1+(−0.5−0.866i)T2 1 + (-0.5 - 0.866i)T^{2}
23 1+(−0.866+0.5i)T2 1 + (-0.866 + 0.5i)T^{2}
29 1+T2 1 + T^{2}
31 1+(1.5−0.866i)T+(0.5−0.866i)T2 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2}
37 1+(−0.448−1.67i)T+(−0.866+0.5i)T2 1 + (-0.448 - 1.67i)T + (-0.866 + 0.5i)T^{2}
41 1+T2 1 + T^{2}
43 1+(1.22+1.22i)T+iT2 1 + (1.22 + 1.22i)T + iT^{2}
47 1+(−0.866+0.5i)T2 1 + (-0.866 + 0.5i)T^{2}
53 1+(0.866+0.5i)T2 1 + (0.866 + 0.5i)T^{2}
59 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
61 1+(−1.5−0.866i)T+(0.5+0.866i)T2 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2}
67 1+(0.866+0.5i)T2 1 + (0.866 + 0.5i)T^{2}
71 1−T2 1 - T^{2}
73 1+(−0.965−0.258i)T+(0.866+0.5i)T2 1 + (-0.965 - 0.258i)T + (0.866 + 0.5i)T^{2}
79 1+(−0.866−0.5i)T+(0.5+0.866i)T2 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2}
83 1+iT2 1 + iT^{2}
89 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
97 1+(−0.707−0.707i)T+iT2 1 + (-0.707 - 0.707i)T + iT^{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

−9.519001126053961240296717318351, −8.740157875364433313411169630441, −7.79818427497818660301527016478, −7.00184892370815675734078463505, −6.45910989544835669009763139893, −5.22989982985694699514846896103, −5.02162789247560460614162068322, −3.97480310746387872597649415265, −2.39614075720857589713401689299, −1.59210099542109706442567642959, 0.55337398647452010738487467046, 2.03757011849520878481465676673, 3.45327380785973644971106557594, 4.40422622272978833381476674233, 5.17238138674655012458035596247, 5.70100850848843929816607374033, 6.84216261333754719761662618008, 7.52900494861069143278501816541, 8.071103083110663958378292077307, 9.397369009358246004137965924135

Graph of the ZZ-function along the critical line