Properties

Label 2-1152-8.5-c3-0-5
Degree 22
Conductor 11521152
Sign 0.7070.707i-0.707 - 0.707i
Analytic cond. 67.970267.9702
Root an. cond. 8.244408.24440
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  + 6i·5-s − 21.1·7-s − 42.3i·11-s + 20i·13-s + 8·17-s − 84.6i·19-s + 169.·23-s + 89·25-s + 46i·29-s + 21.1·31-s − 126. i·35-s + 164i·37-s − 312·41-s + 423. i·43-s − 169.·47-s + ⋯
L(s)  = 1  + 0.536i·5-s − 1.14·7-s − 1.16i·11-s + 0.426i·13-s + 0.114·17-s − 1.02i·19-s + 1.53·23-s + 0.711·25-s + 0.294i·29-s + 0.122·31-s − 0.613i·35-s + 0.728i·37-s − 1.18·41-s + 1.50i·43-s − 0.525·47-s + ⋯

Functional equation

Λ(s)=(1152s/2ΓC(s)L(s)=((0.7070.707i)Λ(4s)\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(1152s/2ΓC(s+3/2)L(s)=((0.7070.707i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 11521152    =    27322^{7} \cdot 3^{2}
Sign: 0.7070.707i-0.707 - 0.707i
Analytic conductor: 67.970267.9702
Root analytic conductor: 8.244408.24440
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ1152(577,)\chi_{1152} (577, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1152, ( :3/2), 0.7070.707i)(2,\ 1152,\ (\ :3/2),\ -0.707 - 0.707i)

Particular Values

L(2)L(2) \approx 0.66912295400.6691229540
L(12)L(\frac12) \approx 0.66912295400.6691229540
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
3 1 1
good5 16iT125T2 1 - 6iT - 125T^{2}
7 1+21.1T+343T2 1 + 21.1T + 343T^{2}
11 1+42.3iT1.33e3T2 1 + 42.3iT - 1.33e3T^{2}
13 120iT2.19e3T2 1 - 20iT - 2.19e3T^{2}
17 18T+4.91e3T2 1 - 8T + 4.91e3T^{2}
19 1+84.6iT6.85e3T2 1 + 84.6iT - 6.85e3T^{2}
23 1169.T+1.21e4T2 1 - 169.T + 1.21e4T^{2}
29 146iT2.43e4T2 1 - 46iT - 2.43e4T^{2}
31 121.1T+2.97e4T2 1 - 21.1T + 2.97e4T^{2}
37 1164iT5.06e4T2 1 - 164iT - 5.06e4T^{2}
41 1+312T+6.89e4T2 1 + 312T + 6.89e4T^{2}
43 1423.iT7.95e4T2 1 - 423. iT - 7.95e4T^{2}
47 1+169.T+1.03e5T2 1 + 169.T + 1.03e5T^{2}
53 1266iT1.48e5T2 1 - 266iT - 1.48e5T^{2}
59 1253.iT2.05e5T2 1 - 253. iT - 2.05e5T^{2}
61 1+132iT2.26e5T2 1 + 132iT - 2.26e5T^{2}
67 1+507.iT3.00e5T2 1 + 507. iT - 3.00e5T^{2}
71 1+677.T+3.57e5T2 1 + 677.T + 3.57e5T^{2}
73 1+246T+3.89e5T2 1 + 246T + 3.89e5T^{2}
79 1+232.T+4.93e5T2 1 + 232.T + 4.93e5T^{2}
83 1973.iT5.71e5T2 1 - 973. iT - 5.71e5T^{2}
89 1+1.39e3T+7.04e5T2 1 + 1.39e3T + 7.04e5T^{2}
97 1+302T+9.12e5T2 1 + 302T + 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

−9.612584530081858222026007341083, −9.028766306522555490089514479270, −8.180281603052474915530171895011, −6.82726479786059202067521158133, −6.70452409333778366591594433586, −5.59942407148581592771416956372, −4.54520414980913956935308001048, −3.11875767581333603192759740542, −2.96907258457680943611021892785, −1.10105881699621432751345220702, 0.17824465339069045753305772108, 1.50701856831571009167156852478, 2.82864035433359919397687291548, 3.77781344438274703600685602465, 4.85134172251848842998253374235, 5.65347616855220775776091121568, 6.73447169633712631361350903956, 7.30918388910267536820033015636, 8.423770315352353201425632992008, 9.155976428761821389169852970025

Graph of the ZZ-function along the critical line