Properties

Label 2-4000-5.4-c1-0-50
Degree 22
Conductor 40004000
Sign 11
Analytic cond. 31.940131.9401
Root an. cond. 5.651565.65156
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  + 0.306i·3-s − 2.60i·7-s + 2.90·9-s + 2.71·11-s − 0.503i·13-s + 5.94i·17-s + 5.72·19-s + 0.799·21-s − 1.28i·23-s + 1.81i·27-s − 2.11·29-s + 3.95·31-s + 0.831i·33-s + 0.825i·37-s + 0.154·39-s + ⋯
L(s)  = 1  + 0.176i·3-s − 0.985i·7-s + 0.968·9-s + 0.818·11-s − 0.139i·13-s + 1.44i·17-s + 1.31·19-s + 0.174·21-s − 0.268i·23-s + 0.348i·27-s − 0.392·29-s + 0.710·31-s + 0.144i·33-s + 0.135i·37-s + 0.0246·39-s + ⋯

Functional equation

Λ(s)=(4000s/2ΓC(s)L(s)=(Λ(2s)\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(4000s/2ΓC(s+1/2)L(s)=(Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 40004000    =    25532^{5} \cdot 5^{3}
Sign: 11
Analytic conductor: 31.940131.9401
Root analytic conductor: 5.651565.65156
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ4000(1249,)\chi_{4000} (1249, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 4000, ( :1/2), 1)(2,\ 4000,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) \approx 2.3413343682.341334368
L(12)L(\frac12) \approx 2.3413343682.341334368
L(32)L(\frac{3}{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
good3 10.306iT3T2 1 - 0.306iT - 3T^{2}
7 1+2.60iT7T2 1 + 2.60iT - 7T^{2}
11 12.71T+11T2 1 - 2.71T + 11T^{2}
13 1+0.503iT13T2 1 + 0.503iT - 13T^{2}
17 15.94iT17T2 1 - 5.94iT - 17T^{2}
19 15.72T+19T2 1 - 5.72T + 19T^{2}
23 1+1.28iT23T2 1 + 1.28iT - 23T^{2}
29 1+2.11T+29T2 1 + 2.11T + 29T^{2}
31 13.95T+31T2 1 - 3.95T + 31T^{2}
37 10.825iT37T2 1 - 0.825iT - 37T^{2}
41 1+4.53T+41T2 1 + 4.53T + 41T^{2}
43 15.38iT43T2 1 - 5.38iT - 43T^{2}
47 15.62iT47T2 1 - 5.62iT - 47T^{2}
53 1+10.9iT53T2 1 + 10.9iT - 53T^{2}
59 113.7T+59T2 1 - 13.7T + 59T^{2}
61 1+7.00T+61T2 1 + 7.00T + 61T^{2}
67 12.85iT67T2 1 - 2.85iT - 67T^{2}
71 1+11.3T+71T2 1 + 11.3T + 71T^{2}
73 110.1iT73T2 1 - 10.1iT - 73T^{2}
79 111.1T+79T2 1 - 11.1T + 79T^{2}
83 1+5.83iT83T2 1 + 5.83iT - 83T^{2}
89 1+13.7T+89T2 1 + 13.7T + 89T^{2}
97 1+7.02iT97T2 1 + 7.02iT - 97T^{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

−8.372969162505242410560318921969, −7.67164165729430461276154082386, −6.97639971187206771505235934338, −6.40396303678978662247444718085, −5.44473799146764295973591890567, −4.44928356940705091490909803987, −3.95298573379079286218799589903, −3.19689926592584631656563085779, −1.71107400391760168829292155130, −0.974295861166339251123190171086, 0.915256463997868912796970581395, 1.94516425331264553051481797436, 2.91203111632549802402398799734, 3.81524129493345317976325938946, 4.78937250188975092036891343517, 5.42842211325491932501383546643, 6.27867802716520724464260757809, 7.12983332746628057564067134505, 7.49405381665830138392612920156, 8.566918253227958342263826728344

Graph of the ZZ-function along the critical line