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Properties

Label 2-1100-5.4-c5-0-19
Degree 22
Conductor 11001100
Sign −0.447−0.894i-0.447 - 0.894i
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  + 7i·3-s + 50i·7-s + 194·9-s + 121·11-s − 380i·13-s + 1.15e3i·17-s + 1.82e3·19-s − 350·21-s + 3.59e3i·23-s + 3.05e3i·27-s − 8.03e3·29-s − 2.94e3·31-s + 847i·33-s − 6.97e3i·37-s + 2.66e3·39-s + ⋯
L(s)  = 1  + 0.449i·3-s + 0.385i·7-s + 0.798·9-s + 0.301·11-s − 0.623i·13-s + 0.968i·17-s + 1.15·19-s − 0.173·21-s + 1.41i·23-s + 0.807i·27-s − 1.77·29-s − 0.550·31-s + 0.135i·33-s − 0.838i·37-s + 0.280·39-s + ⋯

Functional equation

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

Particular Values

L(3)L(3) ≈\approx 2.1150576852.115057685
L(12)L(\frac12) ≈\approx 2.1150576852.115057685
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 1−7iT−243T2 1 - 7iT - 243T^{2}
7 1−50iT−1.68e4T2 1 - 50iT - 1.68e4T^{2}
13 1+380iT−3.71e5T2 1 + 380iT - 3.71e5T^{2}
17 1−1.15e3iT−1.41e6T2 1 - 1.15e3iT - 1.41e6T^{2}
19 1−1.82e3T+2.47e6T2 1 - 1.82e3T + 2.47e6T^{2}
23 1−3.59e3iT−6.43e6T2 1 - 3.59e3iT - 6.43e6T^{2}
29 1+8.03e3T+2.05e7T2 1 + 8.03e3T + 2.05e7T^{2}
31 1+2.94e3T+2.86e7T2 1 + 2.94e3T + 2.86e7T^{2}
37 1+6.97e3iT−6.93e7T2 1 + 6.97e3iT - 6.93e7T^{2}
41 1+520T+1.15e8T2 1 + 520T + 1.15e8T^{2}
43 1+2.48e3iT−1.47e8T2 1 + 2.48e3iT - 1.47e8T^{2}
47 1−6.92e3iT−2.29e8T2 1 - 6.92e3iT - 2.29e8T^{2}
53 1+1.37e4iT−4.18e8T2 1 + 1.37e4iT - 4.18e8T^{2}
59 1−3.17e4T+7.14e8T2 1 - 3.17e4T + 7.14e8T^{2}
61 1−3.41e4T+8.44e8T2 1 - 3.41e4T + 8.44e8T^{2}
67 1−6.15e4iT−1.35e9T2 1 - 6.15e4iT - 1.35e9T^{2}
71 1+1.49e4T+1.80e9T2 1 + 1.49e4T + 1.80e9T^{2}
73 1+3.64e4iT−2.07e9T2 1 + 3.64e4iT - 2.07e9T^{2}
79 1−2.85e4T+3.07e9T2 1 - 2.85e4T + 3.07e9T^{2}
83 1−7.74e4iT−3.93e9T2 1 - 7.74e4iT - 3.93e9T^{2}
89 1+3.62e4T+5.58e9T2 1 + 3.62e4T + 5.58e9T^{2}
97 1−4.97e4iT−8.58e9T2 1 - 4.97e4iT - 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

−9.522555311178225235240628505380, −8.708344793092519895417178738007, −7.63125761581393574153560355830, −7.08754443631387839601495574807, −5.72518767860582419641585187946, −5.32765824077635548425516347533, −3.97351512405805675860523902094, −3.47842994481973804572668659036, −2.04793364656800932560014748981, −1.08434319231107670095343860710, 0.42611698236531113692811863463, 1.35808723449671520236525505996, 2.36835243554766557255993276927, 3.62472445142636645280562348991, 4.48033221431959045235411847088, 5.43262054418980756884997467348, 6.61232539777846507559541582022, 7.15273740425347729425782677851, 7.82610667234566693887726709428, 8.971761369428194800211218926765

Graph of the ZZ-function along the critical line