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Properties

Label 2-4800-5.4-c1-0-62
Degree 22
Conductor 48004800
Sign −0.894−0.447i-0.894 - 0.447i
Analytic cond. 38.328138.3281
Root an. cond. 6.190976.19097
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − i·3-s − 9-s − 4·11-s + 6i·13-s − 6i·17-s + 4·19-s + i·27-s − 2·29-s + 8·31-s + 4i·33-s + 2i·37-s + 6·39-s − 6·41-s − 12i·43-s − 8i·47-s + ⋯
L(s)  = 1  − 0.577i·3-s − 0.333·9-s − 1.20·11-s + 1.66i·13-s − 1.45i·17-s + 0.917·19-s + 0.192i·27-s − 0.371·29-s + 1.43·31-s + 0.696i·33-s + 0.328i·37-s + 0.960·39-s − 0.937·41-s − 1.82i·43-s − 1.16i·47-s + ⋯

Functional equation

Λ(s)=(4800s/2 ΓC(s) L(s)=((−0.894−0.447i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 4800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(4800s/2 ΓC(s+1/2) L(s)=((−0.894−0.447i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 4800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 48004800    =    26⋅3⋅522^{6} \cdot 3 \cdot 5^{2}
Sign: −0.894−0.447i-0.894 - 0.447i
Analytic conductor: 38.328138.3281
Root analytic conductor: 6.190976.19097
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ4800(3649,⋅)\chi_{4800} (3649, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 11
Selberg data: (2, 4800, ( :1/2), −0.894−0.447i)(2,\ 4800,\ (\ :1/2),\ -0.894 - 0.447i)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
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+iT 1 + iT
5 1 1
good7 1−7T2 1 - 7T^{2}
11 1+4T+11T2 1 + 4T + 11T^{2}
13 1−6iT−13T2 1 - 6iT - 13T^{2}
17 1+6iT−17T2 1 + 6iT - 17T^{2}
19 1−4T+19T2 1 - 4T + 19T^{2}
23 1−23T2 1 - 23T^{2}
29 1+2T+29T2 1 + 2T + 29T^{2}
31 1−8T+31T2 1 - 8T + 31T^{2}
37 1−2iT−37T2 1 - 2iT - 37T^{2}
41 1+6T+41T2 1 + 6T + 41T^{2}
43 1+12iT−43T2 1 + 12iT - 43T^{2}
47 1+8iT−47T2 1 + 8iT - 47T^{2}
53 1−6iT−53T2 1 - 6iT - 53T^{2}
59 1+12T+59T2 1 + 12T + 59T^{2}
61 1+14T+61T2 1 + 14T + 61T^{2}
67 1−4iT−67T2 1 - 4iT - 67T^{2}
71 1+8T+71T2 1 + 8T + 71T^{2}
73 1−6iT−73T2 1 - 6iT - 73T^{2}
79 1+8T+79T2 1 + 8T + 79T^{2}
83 1−12iT−83T2 1 - 12iT - 83T^{2}
89 1+10T+89T2 1 + 10T + 89T^{2}
97 1−2iT−97T2 1 - 2iT - 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

−7.68505171874222637609371384280, −7.17901479788646888104313576789, −6.62259967867106113152371797126, −5.61157379274572440855946687444, −5.00983504460379356515178401229, −4.19232013327863941784565934097, −3.01249292175252128478910213895, −2.38621067497179226259211787210, −1.32258944449181823643409938953, 0, 1.40020740604492007621840783555, 2.86035553878473070309564466885, 3.15882638287324621318957699396, 4.31772113101396600190821725962, 5.03694257301407734999652051961, 5.75730889449406377018777755221, 6.26051014659049839744892631401, 7.61845735190994045034698198293, 7.904187680943919217362116464910

Graph of the ZZ-function along the critical line