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Properties

Label 2-1100-55.54-c0-0-1
Degree 22
Conductor 11001100
Sign 0.447+0.894i0.447 + 0.894i
Analytic cond. 0.5489710.548971
Root an. cond. 0.7409260.740926
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  − i·3-s + 11-s − i·23-s − i·27-s − 31-s − i·33-s + i·37-s − 2i·47-s − 49-s + 2i·53-s + 59-s + i·67-s − 69-s − 71-s − 81-s + ⋯
L(s)  = 1  − i·3-s + 11-s − i·23-s − i·27-s − 31-s − i·33-s + i·37-s − 2i·47-s − 49-s + 2i·53-s + 59-s + i·67-s − 69-s − 71-s − 81-s + ⋯

Functional equation

Λ(s)=(1100s/2 ΓC(s) L(s)=((0.447+0.894i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(1100s/2 ΓC(s) L(s)=((0.447+0.894i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 11001100    =    22⋅52⋅112^{2} \cdot 5^{2} \cdot 11
Sign: 0.447+0.894i0.447 + 0.894i
Analytic conductor: 0.5489710.548971
Root analytic conductor: 0.7409260.740926
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ1100(549,⋅)\chi_{1100} (549, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1100, ( :0), 0.447+0.894i)(2,\ 1100,\ (\ :0),\ 0.447 + 0.894i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.1114368741.111436874
L(12)L(\frac12) ≈\approx 1.1114368741.111436874
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
5 1 1
11 1−T 1 - T
good3 1+iT−T2 1 + iT - T^{2}
7 1+T2 1 + T^{2}
13 1+T2 1 + T^{2}
17 1+T2 1 + T^{2}
19 1−T2 1 - T^{2}
23 1+iT−T2 1 + iT - T^{2}
29 1−T2 1 - T^{2}
31 1+T+T2 1 + T + T^{2}
37 1−iT−T2 1 - iT - T^{2}
41 1−T2 1 - T^{2}
43 1+T2 1 + T^{2}
47 1+2iT−T2 1 + 2iT - T^{2}
53 1−2iT−T2 1 - 2iT - T^{2}
59 1−T+T2 1 - T + T^{2}
61 1−T2 1 - T^{2}
67 1−iT−T2 1 - iT - T^{2}
71 1+T+T2 1 + T + T^{2}
73 1+T2 1 + T^{2}
79 1−T2 1 - T^{2}
83 1+T2 1 + T^{2}
89 1−T+T2 1 - T + T^{2}
97 1−iT−T2 1 - iT - 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

−9.894599890739621316269771540695, −8.955274156714402372889349871041, −8.229645119295682048600639656940, −7.25894910792280318490397713639, −6.69872116812168946447000357241, −5.91626090425666631380443876080, −4.68089616261762562149493606104, −3.67034195666631910125126063237, −2.31505650139177241299580054997, −1.21746501681868053595962199274, 1.67903571971531424131634441327, 3.32439945773914004786896152621, 4.02002329170887158538414860367, 4.91740983068476997795306517116, 5.84767285656271670613683837042, 6.86796836654003673273296408697, 7.72249954366541900864644584735, 8.867059245755255398454128152744, 9.446121537586323586289723938045, 10.03140269962709176053957188551

Graph of the ZZ-function along the critical line