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Properties

Label 2-1100-5.4-c5-0-52
Degree 22
Conductor 11001100
Sign 0.894+0.447i0.894 + 0.447i
Analytic cond. 176.422176.422
Root an. cond. 13.282413.2824
Motivic weight 55
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 26.6i·3-s + 210. i·7-s − 466.·9-s + 121·11-s − 457. i·13-s + 531. i·17-s − 515.·19-s − 5.61e3·21-s + 2.84e3i·23-s − 5.94e3i·27-s + 443.·29-s − 3.97e3·31-s + 3.22e3i·33-s − 1.31e4i·37-s + 1.21e4·39-s + ⋯
L(s)  = 1  + 1.70i·3-s + 1.62i·7-s − 1.91·9-s + 0.301·11-s − 0.750i·13-s + 0.446i·17-s − 0.327·19-s − 2.77·21-s + 1.12i·23-s − 1.56i·27-s + 0.0978·29-s − 0.742·31-s + 0.515i·33-s − 1.58i·37-s + 1.28·39-s + ⋯

Functional equation

Λ(s)=(1100s/2 ΓC(s) L(s)=((0.894+0.447i) Λ‾(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(6-s) \end{aligned}
Λ(s)=(1100s/2 ΓC(s+5/2) L(s)=((0.894+0.447i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 11001100    =    22⋅52⋅112^{2} \cdot 5^{2} \cdot 11
Sign: 0.894+0.447i0.894 + 0.447i
Analytic conductor: 176.422176.422
Root analytic conductor: 13.282413.2824
Motivic weight: 55
Rational: no
Arithmetic: yes
Character: χ1100(749,⋅)\chi_{1100} (749, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1100, ( :5/2), 0.894+0.447i)(2,\ 1100,\ (\ :5/2),\ 0.894 + 0.447i)

Particular Values

L(3)L(3) ≈\approx 0.12803678040.1280367804
L(12)L(\frac12) ≈\approx 0.12803678040.1280367804
L(72)L(\frac{7}{2}) 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 1−121T 1 - 121T
good3 1−26.6iT−243T2 1 - 26.6iT - 243T^{2}
7 1−210.iT−1.68e4T2 1 - 210. iT - 1.68e4T^{2}
13 1+457.iT−3.71e5T2 1 + 457. iT - 3.71e5T^{2}
17 1−531.iT−1.41e6T2 1 - 531. iT - 1.41e6T^{2}
19 1+515.T+2.47e6T2 1 + 515.T + 2.47e6T^{2}
23 1−2.84e3iT−6.43e6T2 1 - 2.84e3iT - 6.43e6T^{2}
29 1−443.T+2.05e7T2 1 - 443.T + 2.05e7T^{2}
31 1+3.97e3T+2.86e7T2 1 + 3.97e3T + 2.86e7T^{2}
37 1+1.31e4iT−6.93e7T2 1 + 1.31e4iT - 6.93e7T^{2}
41 1+530.T+1.15e8T2 1 + 530.T + 1.15e8T^{2}
43 1+1.91e3iT−1.47e8T2 1 + 1.91e3iT - 1.47e8T^{2}
47 1+1.33e4iT−2.29e8T2 1 + 1.33e4iT - 2.29e8T^{2}
53 1−1.93e4iT−4.18e8T2 1 - 1.93e4iT - 4.18e8T^{2}
59 1+3.15e4T+7.14e8T2 1 + 3.15e4T + 7.14e8T^{2}
61 1+1.20e4T+8.44e8T2 1 + 1.20e4T + 8.44e8T^{2}
67 1+3.15e4iT−1.35e9T2 1 + 3.15e4iT - 1.35e9T^{2}
71 1+9.16e3T+1.80e9T2 1 + 9.16e3T + 1.80e9T^{2}
73 1+2.69e4iT−2.07e9T2 1 + 2.69e4iT - 2.07e9T^{2}
79 1−6.54e4T+3.07e9T2 1 - 6.54e4T + 3.07e9T^{2}
83 1+4.71e4iT−3.93e9T2 1 + 4.71e4iT - 3.93e9T^{2}
89 1+2.77e4T+5.58e9T2 1 + 2.77e4T + 5.58e9T^{2}
97 1+1.33e5iT−8.58e9T2 1 + 1.33e5iT - 8.58e9T^{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

−9.087096904233841052361777486352, −8.696333613990979224310694057391, −7.63356753734935064382727355613, −6.04809029971237095921209714073, −5.58554366271268116454335178780, −4.82905120711980314701645781314, −3.76201402131411278617520671787, −3.04670700992558287017569500125, −1.95640564894760578157892667047, −0.02650031426718303801556754171, 0.908241317144951174689116027231, 1.58119526631057442314114709293, 2.69570493109021607004450118214, 3.90336460812283686085735858350, 4.88497996730206777083167695478, 6.37980913655533982762595945666, 6.67803661739543867059495580651, 7.44815736038680089766635168503, 8.063473864004356841697414348360, 8.986214886033112834955706483253

Graph of the ZZ-function along the critical line