Properties

Label 2-6000-5.4-c1-0-45
Degree 22
Conductor 60006000
Sign i-i
Analytic cond. 47.910247.9102
Root an. cond. 6.921726.92172
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 + 1.38i·7-s − 9-s + 6.23·11-s + 3i·13-s − 3.38i·17-s − 19-s − 1.38·21-s + 6.70i·23-s i·27-s + 4.23·29-s + 3.09·31-s + 6.23i·33-s − 5i·37-s − 3·39-s + ⋯
L(s)  = 1  + 0.577i·3-s + 0.522i·7-s − 0.333·9-s + 1.88·11-s + 0.832i·13-s − 0.820i·17-s − 0.229·19-s − 0.301·21-s + 1.39i·23-s − 0.192i·27-s + 0.786·29-s + 0.555·31-s + 1.08i·33-s − 0.821i·37-s − 0.480·39-s + ⋯

Functional equation

Λ(s)=(6000s/2ΓC(s)L(s)=(iΛ(2s)\begin{aligned}\Lambda(s)=\mathstrut & 6000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(6000s/2ΓC(s+1/2)L(s)=(iΛ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 6000 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 60006000    =    243532^{4} \cdot 3 \cdot 5^{3}
Sign: i-i
Analytic conductor: 47.910247.9102
Root analytic conductor: 6.921726.92172
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ6000(1249,)\chi_{6000} (1249, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 6000, ( :1/2), i)(2,\ 6000,\ (\ :1/2),\ -i)

Particular Values

L(1)L(1) \approx 2.1948721072.194872107
L(12)L(\frac12) \approx 2.1948721072.194872107
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
3 1iT 1 - iT
5 1 1
good7 11.38iT7T2 1 - 1.38iT - 7T^{2}
11 16.23T+11T2 1 - 6.23T + 11T^{2}
13 13iT13T2 1 - 3iT - 13T^{2}
17 1+3.38iT17T2 1 + 3.38iT - 17T^{2}
19 1+T+19T2 1 + T + 19T^{2}
23 16.70iT23T2 1 - 6.70iT - 23T^{2}
29 14.23T+29T2 1 - 4.23T + 29T^{2}
31 13.09T+31T2 1 - 3.09T + 31T^{2}
37 1+5iT37T2 1 + 5iT - 37T^{2}
41 1+4.14T+41T2 1 + 4.14T + 41T^{2}
43 12.38iT43T2 1 - 2.38iT - 43T^{2}
47 19.18iT47T2 1 - 9.18iT - 47T^{2}
53 1+3.61iT53T2 1 + 3.61iT - 53T^{2}
59 110.7T+59T2 1 - 10.7T + 59T^{2}
61 1+5.09T+61T2 1 + 5.09T + 61T^{2}
67 1+8iT67T2 1 + 8iT - 67T^{2}
71 1+1.14T+71T2 1 + 1.14T + 71T^{2}
73 19.14iT73T2 1 - 9.14iT - 73T^{2}
79 12.76T+79T2 1 - 2.76T + 79T^{2}
83 18.32iT83T2 1 - 8.32iT - 83T^{2}
89 15.47T+89T2 1 - 5.47T + 89T^{2}
97 1+9.56iT97T2 1 + 9.56iT - 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

−8.503319696765131294705433582317, −7.47974732061360613195214676689, −6.73908267041762519781100356696, −6.18130475766725826402720921661, −5.36166389742219699620843249776, −4.52404177533123052086773549903, −3.92770556282587765399507964437, −3.14516942269596473678648841214, −2.08903039400334965441480885432, −1.09247224217868012486904249346, 0.66049535098858556224487652619, 1.43505483538762821959349875866, 2.48481502600940957506789644544, 3.52634199587743765933659664259, 4.13977230156081707623358722189, 4.98242306379367774270142354107, 6.06864532716992106797738150033, 6.51979606830412014128565081458, 7.03724175706636410580252218956, 7.952537303851671843929230123238

Graph of the ZZ-function along the critical line