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Properties

Label 2-1920-8.5-c1-0-24
Degree 22
Conductor 19201920
Sign −0.707+0.707i-0.707 + 0.707i
Analytic cond. 15.331215.3312
Root an. cond. 3.915513.91551
Motivic weight 11
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 + i·5-s − 2·7-s − 9-s + 2i·11-s − 2i·13-s + 15-s + 4·17-s − 4i·19-s + 2i·21-s − 4·23-s − 25-s + i·27-s − 2i·29-s − 4·31-s + ⋯
L(s)  = 1  − 0.577i·3-s + 0.447i·5-s − 0.755·7-s − 0.333·9-s + 0.603i·11-s − 0.554i·13-s + 0.258·15-s + 0.970·17-s − 0.917i·19-s + 0.436i·21-s − 0.834·23-s − 0.200·25-s + 0.192i·27-s − 0.371i·29-s − 0.718·31-s + ⋯

Functional equation

Λ(s)=(1920s/2 ΓC(s) L(s)=((−0.707+0.707i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1920s/2 ΓC(s+1/2) L(s)=((−0.707+0.707i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 19201920    =    27⋅3⋅52^{7} \cdot 3 \cdot 5
Sign: −0.707+0.707i-0.707 + 0.707i
Analytic conductor: 15.331215.3312
Root analytic conductor: 3.915513.91551
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1920(961,⋅)\chi_{1920} (961, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1920, ( :1/2), −0.707+0.707i)(2,\ 1920,\ (\ :1/2),\ -0.707 + 0.707i)

Particular Values

L(1)L(1) ≈\approx 0.78889458460.7888945846
L(12)L(\frac12) ≈\approx 0.78889458460.7888945846
L(32)L(\frac{3}{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
3 1+iT 1 + iT
5 1−iT 1 - iT
good7 1+2T+7T2 1 + 2T + 7T^{2}
11 1−2iT−11T2 1 - 2iT - 11T^{2}
13 1+2iT−13T2 1 + 2iT - 13T^{2}
17 1−4T+17T2 1 - 4T + 17T^{2}
19 1+4iT−19T2 1 + 4iT - 19T^{2}
23 1+4T+23T2 1 + 4T + 23T^{2}
29 1+2iT−29T2 1 + 2iT - 29T^{2}
31 1+4T+31T2 1 + 4T + 31T^{2}
37 1+2iT−37T2 1 + 2iT - 37T^{2}
41 1−6T+41T2 1 - 6T + 41T^{2}
43 1+4iT−43T2 1 + 4iT - 43T^{2}
47 1+8T+47T2 1 + 8T + 47T^{2}
53 1+10iT−53T2 1 + 10iT - 53T^{2}
59 1+6iT−59T2 1 + 6iT - 59T^{2}
61 1−61T2 1 - 61T^{2}
67 1+12iT−67T2 1 + 12iT - 67T^{2}
71 1+8T+71T2 1 + 8T + 71T^{2}
73 1+6T+73T2 1 + 6T + 73T^{2}
79 1+4T+79T2 1 + 4T + 79T^{2}
83 1+16iT−83T2 1 + 16iT - 83T^{2}
89 1+6T+89T2 1 + 6T + 89T^{2}
97 1+14T+97T2 1 + 14T + 97T^{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

−8.937392507723227211213638940724, −7.899202937869052609940500356239, −7.37914790000760467276501922256, −6.55596640815945222061463406359, −5.89309596563536002351928267567, −4.94519115195961000641508632514, −3.69469712571173470165046296228, −2.91086673768889109528669180180, −1.86768577791447890582072157859, −0.28752645143261348999817777872, 1.38644709173511764815421094461, 2.88724812615568754402538124865, 3.71639244827035964013692630335, 4.48796454073255759835434882804, 5.66396393363963465040847778028, 6.03297402500341719169067061997, 7.17606597733154616563819357780, 8.091336129868759217900724226901, 8.767485436494887063419299969576, 9.641214052552447118854817027544

Graph of the ZZ-function along the critical line