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Properties

Label 2-2400-40.29-c1-0-28
Degree 22
Conductor 24002400
Sign 0.606+0.794i0.606 + 0.794i
Analytic cond. 19.164019.1640
Root an. cond. 4.377684.37768
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  + 3-s + 0.941i·7-s + 9-s − 4.49i·11-s + 5.55·13-s − 7.55i·17-s + 1.05i·19-s + 0.941i·21-s + 1.05i·23-s + 27-s + 2i·29-s − 3.55·31-s − 4.49i·33-s − 7.43·37-s + 5.55·39-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.355i·7-s + 0.333·9-s − 1.35i·11-s + 1.54·13-s − 1.83i·17-s + 0.242i·19-s + 0.205i·21-s + 0.220i·23-s + 0.192·27-s + 0.371i·29-s − 0.638·31-s − 0.783i·33-s − 1.22·37-s + 0.889·39-s + ⋯

Functional equation

Λ(s)=(2400s/2 ΓC(s) L(s)=((0.606+0.794i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.606 + 0.794i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(2400s/2 ΓC(s+1/2) L(s)=((0.606+0.794i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.606 + 0.794i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 24002400    =    25⋅3⋅522^{5} \cdot 3 \cdot 5^{2}
Sign: 0.606+0.794i0.606 + 0.794i
Analytic conductor: 19.164019.1640
Root analytic conductor: 4.377684.37768
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ2400(49,⋅)\chi_{2400} (49, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2400, ( :1/2), 0.606+0.794i)(2,\ 2400,\ (\ :1/2),\ 0.606 + 0.794i)

Particular Values

L(1)L(1) ≈\approx 2.2631615582.263161558
L(12)L(\frac12) ≈\approx 2.2631615582.263161558
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−T 1 - T
5 1 1
good7 1−0.941iT−7T2 1 - 0.941iT - 7T^{2}
11 1+4.49iT−11T2 1 + 4.49iT - 11T^{2}
13 1−5.55T+13T2 1 - 5.55T + 13T^{2}
17 1+7.55iT−17T2 1 + 7.55iT - 17T^{2}
19 1−1.05iT−19T2 1 - 1.05iT - 19T^{2}
23 1−1.05iT−23T2 1 - 1.05iT - 23T^{2}
29 1−2iT−29T2 1 - 2iT - 29T^{2}
31 1+3.55T+31T2 1 + 3.55T + 31T^{2}
37 1+7.43T+37T2 1 + 7.43T + 37T^{2}
41 1+3.88T+41T2 1 + 3.88T + 41T^{2}
43 1+1.88T+43T2 1 + 1.88T + 43T^{2}
47 1+10.0iT−47T2 1 + 10.0iT - 47T^{2}
53 1+2T+53T2 1 + 2T + 53T^{2}
59 1+8.49iT−59T2 1 + 8.49iT - 59T^{2}
61 1−8.99iT−61T2 1 - 8.99iT - 61T^{2}
67 1−4T+67T2 1 - 4T + 67T^{2}
71 1−12.9T+71T2 1 - 12.9T + 71T^{2}
73 1+6iT−73T2 1 + 6iT - 73T^{2}
79 1−11.5T+79T2 1 - 11.5T + 79T^{2}
83 1−5.88T+83T2 1 - 5.88T + 83T^{2}
89 1−4.11T+89T2 1 - 4.11T + 89T^{2}
97 1+17.1iT−97T2 1 + 17.1iT - 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

−8.738519681500098348332404805979, −8.357130189427668794288567852544, −7.34976509347759265874039864533, −6.57049953000727198155915988785, −5.68303422865833318385383979087, −4.99908492486562611148322788205, −3.58914634324196703489370340258, −3.29770649476703901681287696962, −2.05236991245424410707104141295, −0.75956929801200665649749317828, 1.36528714491591577329387285995, 2.16922650646852252975210900009, 3.58259352866861253422616305858, 4.00636078270904823089089889224, 5.01558600988631020394020957032, 6.14446201947570414896676947317, 6.75390879726032345461048178176, 7.66992977124106132448375118591, 8.318365851761539996503590720279, 8.980833175997487257980876594758

Graph of the ZZ-function along the critical line