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Properties

Label 2-5760-120.59-c1-0-59
Degree 22
Conductor 57605760
Sign 0.988+0.151i0.988 + 0.151i
Analytic cond. 45.993845.9938
Root an. cond. 6.781876.78187
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  + (−0.707 + 2.12i)5-s + 2.82·7-s + 4.24·17-s − 4i·23-s + (−3.99 − 3i)25-s + 4.24·29-s − 8.48i·31-s + (−2.00 + 6i)35-s + 6·37-s − 4.24i·41-s − 5.65i·43-s − 4i·47-s + 1.00·49-s + 4.24i·53-s + 12i·59-s + ⋯
L(s)  = 1  + (−0.316 + 0.948i)5-s + 1.06·7-s + 1.02·17-s − 0.834i·23-s + (−0.799 − 0.600i)25-s + 0.787·29-s − 1.52i·31-s + (−0.338 + 1.01i)35-s + 0.986·37-s − 0.662i·41-s − 0.862i·43-s − 0.583i·47-s + 0.142·49-s + 0.582i·53-s + 1.56i·59-s + ⋯

Functional equation

Λ(s)=(5760s/2 ΓC(s) L(s)=((0.988+0.151i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 5760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.988 + 0.151i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(5760s/2 ΓC(s+1/2) L(s)=((0.988+0.151i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 5760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.988 + 0.151i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 57605760    =    27⋅32⋅52^{7} \cdot 3^{2} \cdot 5
Sign: 0.988+0.151i0.988 + 0.151i
Analytic conductor: 45.993845.9938
Root analytic conductor: 6.781876.78187
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ5760(2879,⋅)\chi_{5760} (2879, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 5760, ( :1/2), 0.988+0.151i)(2,\ 5760,\ (\ :1/2),\ 0.988 + 0.151i)

Particular Values

L(1)L(1) ≈\approx 2.1858697112.185869711
L(12)L(\frac12) ≈\approx 2.1858697112.185869711
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 1
5 1+(0.707−2.12i)T 1 + (0.707 - 2.12i)T
good7 1−2.82T+7T2 1 - 2.82T + 7T^{2}
11 1−11T2 1 - 11T^{2}
13 1+13T2 1 + 13T^{2}
17 1−4.24T+17T2 1 - 4.24T + 17T^{2}
19 1+19T2 1 + 19T^{2}
23 1+4iT−23T2 1 + 4iT - 23T^{2}
29 1−4.24T+29T2 1 - 4.24T + 29T^{2}
31 1+8.48iT−31T2 1 + 8.48iT - 31T^{2}
37 1−6T+37T2 1 - 6T + 37T^{2}
41 1+4.24iT−41T2 1 + 4.24iT - 41T^{2}
43 1+5.65iT−43T2 1 + 5.65iT - 43T^{2}
47 1+4iT−47T2 1 + 4iT - 47T^{2}
53 1−4.24iT−53T2 1 - 4.24iT - 53T^{2}
59 1−12iT−59T2 1 - 12iT - 59T^{2}
61 1+8iT−61T2 1 + 8iT - 61T^{2}
67 1+11.3iT−67T2 1 + 11.3iT - 67T^{2}
71 1−12T+71T2 1 - 12T + 71T^{2}
73 1+6iT−73T2 1 + 6iT - 73T^{2}
79 1+8.48iT−79T2 1 + 8.48iT - 79T^{2}
83 1−4T+83T2 1 - 4T + 83T^{2}
89 1−7.07iT−89T2 1 - 7.07iT - 89T^{2}
97 1−6iT−97T2 1 - 6iT - 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

−7.85439455667584582010179816474, −7.63971438908482009429392962392, −6.68254569661503189081587677878, −6.03003588519368312924565715996, −5.20869512070133951188578082011, −4.38502388096969163717252620122, −3.67934992940711884315638663380, −2.72025137987748086078361590478, −1.97527155561328159304785324036, −0.68448656404734519856781857105, 1.01737622402434434673120839396, 1.57739327321865674109395996583, 2.86168702193150599922765552129, 3.80018469238022702204798418264, 4.63699483908939159227837482055, 5.13880158617508285231346755160, 5.78596217989301688113056473305, 6.80251651685314032528018834640, 7.66125172031772712220923181857, 8.171753994990697508151275186078

Graph of the ZZ-function along the critical line