Properties

Label 2-1100-5.4-c3-0-42
Degree 22
Conductor 11001100
Sign 0.894+0.447i-0.894 + 0.447i
Analytic cond. 64.902164.9021
Root an. cond. 8.056188.05618
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.53i·3-s − 31.0i·7-s + 20.5·9-s + 11·11-s − 79.1i·13-s − 54.4i·17-s − 105.·19-s + 78.8·21-s − 72.0i·23-s + 120. i·27-s − 228.·29-s + 22.7·31-s + 27.9i·33-s − 140. i·37-s + 201.·39-s + ⋯
L(s)  = 1  + 0.488i·3-s − 1.67i·7-s + 0.761·9-s + 0.301·11-s − 1.68i·13-s − 0.776i·17-s − 1.27·19-s + 0.819·21-s − 0.653i·23-s + 0.860i·27-s − 1.46·29-s + 0.132·31-s + 0.147i·33-s − 0.624i·37-s + 0.825·39-s + ⋯

Functional equation

Λ(s)=(1100s/2ΓC(s)L(s)=((0.894+0.447i)Λ(4s)\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(1100s/2ΓC(s+3/2)L(s)=((0.894+0.447i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 11001100    =    2252112^{2} \cdot 5^{2} \cdot 11
Sign: 0.894+0.447i-0.894 + 0.447i
Analytic conductor: 64.902164.9021
Root analytic conductor: 8.056188.05618
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ1100(749,)\chi_{1100} (749, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1100, ( :3/2), 0.894+0.447i)(2,\ 1100,\ (\ :3/2),\ -0.894 + 0.447i)

Particular Values

L(2)L(2) \approx 1.1237758671.123775867
L(12)L(\frac12) \approx 1.1237758671.123775867
L(52)L(\frac{5}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1 1
5 1 1
11 111T 1 - 11T
good3 12.53iT27T2 1 - 2.53iT - 27T^{2}
7 1+31.0iT343T2 1 + 31.0iT - 343T^{2}
13 1+79.1iT2.19e3T2 1 + 79.1iT - 2.19e3T^{2}
17 1+54.4iT4.91e3T2 1 + 54.4iT - 4.91e3T^{2}
19 1+105.T+6.85e3T2 1 + 105.T + 6.85e3T^{2}
23 1+72.0iT1.21e4T2 1 + 72.0iT - 1.21e4T^{2}
29 1+228.T+2.43e4T2 1 + 228.T + 2.43e4T^{2}
31 122.7T+2.97e4T2 1 - 22.7T + 2.97e4T^{2}
37 1+140.iT5.06e4T2 1 + 140. iT - 5.06e4T^{2}
41 162.8T+6.89e4T2 1 - 62.8T + 6.89e4T^{2}
43 1481.iT7.95e4T2 1 - 481. iT - 7.95e4T^{2}
47 169.1iT1.03e5T2 1 - 69.1iT - 1.03e5T^{2}
53 1453.iT1.48e5T2 1 - 453. iT - 1.48e5T^{2}
59 1225.T+2.05e5T2 1 - 225.T + 2.05e5T^{2}
61 1+320.T+2.26e5T2 1 + 320.T + 2.26e5T^{2}
67 1200.iT3.00e5T2 1 - 200. iT - 3.00e5T^{2}
71 1+971.T+3.57e5T2 1 + 971.T + 3.57e5T^{2}
73 1688.iT3.89e5T2 1 - 688. iT - 3.89e5T^{2}
79 1544.T+4.93e5T2 1 - 544.T + 4.93e5T^{2}
83 1788.iT5.71e5T2 1 - 788. iT - 5.71e5T^{2}
89 1+1.11e3T+7.04e5T2 1 + 1.11e3T + 7.04e5T^{2}
97 1842.iT9.12e5T2 1 - 842. iT - 9.12e5T^{2}
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   L(s)=p j=12(1αj,pps)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

−9.367317718825131111281809382743, −8.153275531687693652417557338989, −7.45820211613666968506446685197, −6.77688844808574006176416843998, −5.63725468230956603068156520032, −4.47783794696925704564264749556, −4.02093066872797519393076560317, −2.92475755677558875260601969504, −1.28870530319547676419461106603, −0.26932951511772553823108687709, 1.78695922576799862318158775069, 2.07285914302726225534194578209, 3.69116141940300616569805157187, 4.61964148712638756077949145640, 5.78299584389586882416766145959, 6.45094978440253217498998710487, 7.21439202126780774098295157289, 8.336382963774411364187506061180, 8.989705869271898138073016463000, 9.578165058157278552342524502100

Graph of the ZZ-function along the critical line