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Properties

Label 2-2100-105.68-c0-0-2
Degree 22
Conductor 21002100
Sign 0.910−0.413i0.910 - 0.413i
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 + (0.965 − 0.258i)7-s + (0.866 + 0.499i)9-s + (−0.707 + 0.707i)13-s + (−0.866 + 1.5i)19-s + 21-s + (0.707 + 0.707i)27-s + (1.5 − 0.866i)31-s + (−0.448 − 1.67i)37-s + (−0.866 + 0.500i)39-s + (−1.22 − 1.22i)43-s + (0.866 − 0.499i)49-s + (−1.22 + 1.22i)57-s + (0.965 + 0.258i)63-s + (−1.67 − 0.448i)67-s + ⋯
L(s)  = 1  + (0.965 + 0.258i)3-s + (0.965 − 0.258i)7-s + (0.866 + 0.499i)9-s + (−0.707 + 0.707i)13-s + (−0.866 + 1.5i)19-s + 21-s + (0.707 + 0.707i)27-s + (1.5 − 0.866i)31-s + (−0.448 − 1.67i)37-s + (−0.866 + 0.500i)39-s + (−1.22 − 1.22i)43-s + (0.866 − 0.499i)49-s + (−1.22 + 1.22i)57-s + (0.965 + 0.258i)63-s + (−1.67 − 0.448i)67-s + ⋯

Functional equation

Λ(s)=(2100s/2 ΓC(s) L(s)=((0.910−0.413i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.910 - 0.413i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(2100s/2 ΓC(s) L(s)=((0.910−0.413i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.910 - 0.413i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 21002100    =    22⋅3⋅52⋅72^{2} \cdot 3 \cdot 5^{2} \cdot 7
Sign: 0.910−0.413i0.910 - 0.413i
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), 0.910−0.413i)(2,\ 2100,\ (\ :0),\ 0.910 - 0.413i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.7660553401.766055340
L(12)L(\frac12) ≈\approx 1.7660553401.766055340
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+(−0.965+0.258i)T 1 + (-0.965 + 0.258i)T
good11 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
13 1+(0.707−0.707i)T−iT2 1 + (0.707 - 0.707i)T - iT^{2}
17 1+(0.866+0.5i)T2 1 + (0.866 + 0.5i)T^{2}
19 1+(0.866−1.5i)T+(−0.5−0.866i)T2 1 + (0.866 - 1.5i)T + (-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+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
67 1+(1.67+0.448i)T+(0.866+0.5i)T2 1 + (1.67 + 0.448i)T + (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+(−1.41−1.41i)T+iT2 1 + (-1.41 - 1.41i)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

−9.206487736433943890359698183453, −8.555186686020699933265615916639, −7.85899440264884426271411918653, −7.30521776544419520617471562294, −6.27753049747185965532118923706, −5.13363840062381935359766167564, −4.32969160129989271305096442946, −3.72219632302965796290347689337, −2.38037071433561241916760339144, −1.68580323058487898409502559381, 1.35093327462798419083978319162, 2.50174319384013010363207105439, 3.13449517558234501863118962822, 4.58437513763845363900613186168, 4.88615937348950763466938400600, 6.28630303228383993132731072213, 7.03798931413965310003979527287, 7.86536072808857315033621750856, 8.448415657712571029589290197730, 8.988636256567794050799568312868

Graph of the ZZ-function along the critical line