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Properties

Label 2-1100-5.4-c5-0-15
Degree 22
Conductor 11001100
Sign 0.447−0.894i0.447 - 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  − 11.4i·3-s + 39.4i·7-s + 111.·9-s − 121·11-s + 1.18e3i·13-s − 2.19e3i·17-s − 1.86e3·19-s + 452.·21-s − 337. i·23-s − 4.06e3i·27-s + 5.43e3·29-s + 139.·31-s + 1.38e3i·33-s + 5.94e3i·37-s + 1.35e4·39-s + ⋯
L(s)  = 1  − 0.736i·3-s + 0.303i·7-s + 0.457·9-s − 0.301·11-s + 1.94i·13-s − 1.84i·17-s − 1.18·19-s + 0.223·21-s − 0.133i·23-s − 1.07i·27-s + 1.19·29-s + 0.0260·31-s + 0.222i·33-s + 0.713i·37-s + 1.42·39-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: 0.447−0.894i0.447 - 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 1.4956665401.495666540
L(12)L(\frac12) ≈\approx 1.4956665401.495666540
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+11.4iT−243T2 1 + 11.4iT - 243T^{2}
7 1−39.4iT−1.68e4T2 1 - 39.4iT - 1.68e4T^{2}
13 1−1.18e3iT−3.71e5T2 1 - 1.18e3iT - 3.71e5T^{2}
17 1+2.19e3iT−1.41e6T2 1 + 2.19e3iT - 1.41e6T^{2}
19 1+1.86e3T+2.47e6T2 1 + 1.86e3T + 2.47e6T^{2}
23 1+337.iT−6.43e6T2 1 + 337. iT - 6.43e6T^{2}
29 1−5.43e3T+2.05e7T2 1 - 5.43e3T + 2.05e7T^{2}
31 1−139.T+2.86e7T2 1 - 139.T + 2.86e7T^{2}
37 1−5.94e3iT−6.93e7T2 1 - 5.94e3iT - 6.93e7T^{2}
41 1−4.63e3T+1.15e8T2 1 - 4.63e3T + 1.15e8T^{2}
43 1−1.15e3iT−1.47e8T2 1 - 1.15e3iT - 1.47e8T^{2}
47 1−3.41e3iT−2.29e8T2 1 - 3.41e3iT - 2.29e8T^{2}
53 1+5.65e3iT−4.18e8T2 1 + 5.65e3iT - 4.18e8T^{2}
59 1+1.36e4T+7.14e8T2 1 + 1.36e4T + 7.14e8T^{2}
61 1+3.42e4T+8.44e8T2 1 + 3.42e4T + 8.44e8T^{2}
67 1+9.96e3iT−1.35e9T2 1 + 9.96e3iT - 1.35e9T^{2}
71 1+1.98e4T+1.80e9T2 1 + 1.98e4T + 1.80e9T^{2}
73 1−5.16e4iT−2.07e9T2 1 - 5.16e4iT - 2.07e9T^{2}
79 1+5.09e4T+3.07e9T2 1 + 5.09e4T + 3.07e9T^{2}
83 1−8.62e3iT−3.93e9T2 1 - 8.62e3iT - 3.93e9T^{2}
89 1−2.99e4T+5.58e9T2 1 - 2.99e4T + 5.58e9T^{2}
97 1−4.88e4iT−8.58e9T2 1 - 4.88e4iT - 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

−9.215525648740368380606469758025, −8.475260948014453464421385932999, −7.42723736850743746645716444086, −6.81901904183637951233257218490, −6.19842363994151196010538140998, −4.82462188272964759421427834055, −4.26193077337945109707428935856, −2.73146857605401325412102011246, −1.98420852725155193477700238157, −0.932995029640330184735261941202, 0.31192322055350848010806043173, 1.52626091695872239114320824093, 2.85097396759804637067713243550, 3.82411314157394193909290391569, 4.52089667855864778015685285805, 5.56250393273855770003121064106, 6.30358287705083717695681325308, 7.49371227597519187195961149195, 8.196201185228477294043659081357, 8.946380396210580849151096313167

Graph of the ZZ-function along the critical line