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Properties

Label 2-2400-60.59-c1-0-61
Degree 22
Conductor 24002400
Sign −0.151+0.988i-0.151 + 0.988i
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  + (1.70 − 0.292i)3-s − 3.41·7-s + (2.82 − i)9-s + 2.82·11-s − 2i·13-s − 7.65·17-s + 2.82i·19-s + (−5.82 + i)21-s − 7.41i·23-s + (4.53 − 2.53i)27-s − 8i·29-s − 5.65i·31-s + (4.82 − 0.828i)33-s + 0.343i·37-s + (−0.585 − 3.41i)39-s + ⋯
L(s)  = 1  + (0.985 − 0.169i)3-s − 1.29·7-s + (0.942 − 0.333i)9-s + 0.852·11-s − 0.554i·13-s − 1.85·17-s + 0.648i·19-s + (−1.27 + 0.218i)21-s − 1.54i·23-s + (0.872 − 0.487i)27-s − 1.48i·29-s − 1.01i·31-s + (0.840 − 0.144i)33-s + 0.0564i·37-s + (−0.0938 − 0.546i)39-s + ⋯

Functional equation

Λ(s)=(2400s/2 ΓC(s) L(s)=((−0.151+0.988i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.151 + 0.988i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(2400s/2 ΓC(s+1/2) L(s)=((−0.151+0.988i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.151 + 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 24002400    =    25⋅3⋅522^{5} \cdot 3 \cdot 5^{2}
Sign: −0.151+0.988i-0.151 + 0.988i
Analytic conductor: 19.164019.1640
Root analytic conductor: 4.377684.37768
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ2400(2399,⋅)\chi_{2400} (2399, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2400, ( :1/2), −0.151+0.988i)(2,\ 2400,\ (\ :1/2),\ -0.151 + 0.988i)

Particular Values

L(1)L(1) ≈\approx 1.7427004981.742700498
L(12)L(\frac12) ≈\approx 1.7427004981.742700498
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+(−1.70+0.292i)T 1 + (-1.70 + 0.292i)T
5 1 1
good7 1+3.41T+7T2 1 + 3.41T + 7T^{2}
11 1−2.82T+11T2 1 - 2.82T + 11T^{2}
13 1+2iT−13T2 1 + 2iT - 13T^{2}
17 1+7.65T+17T2 1 + 7.65T + 17T^{2}
19 1−2.82iT−19T2 1 - 2.82iT - 19T^{2}
23 1+7.41iT−23T2 1 + 7.41iT - 23T^{2}
29 1+8iT−29T2 1 + 8iT - 29T^{2}
31 1+5.65iT−31T2 1 + 5.65iT - 31T^{2}
37 1−0.343iT−37T2 1 - 0.343iT - 37T^{2}
41 1+2iT−41T2 1 + 2iT - 41T^{2}
43 1−7.89T+43T2 1 - 7.89T + 43T^{2}
47 1+6.24iT−47T2 1 + 6.24iT - 47T^{2}
53 1+3.65T+53T2 1 + 3.65T + 53T^{2}
59 1−1.65T+59T2 1 - 1.65T + 59T^{2}
61 1−5.65T+61T2 1 - 5.65T + 61T^{2}
67 1−1.07T+67T2 1 - 1.07T + 67T^{2}
71 1+1.17T+71T2 1 + 1.17T + 71T^{2}
73 1−15.6iT−73T2 1 - 15.6iT - 73T^{2}
79 1−4.48iT−79T2 1 - 4.48iT - 79T^{2}
83 1+5.07iT−83T2 1 + 5.07iT - 83T^{2}
89 1+7.31iT−89T2 1 + 7.31iT - 89T^{2}
97 1+18.9iT−97T2 1 + 18.9iT - 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

−8.724883395859655719714803863985, −8.184463292276824541642759071099, −7.10648040450643913356269732932, −6.55669595309024432048309908971, −5.91901269074076452941616561502, −4.31588032579121051688519370677, −3.93885803989304501445664910713, −2.81057384900488330645414989652, −2.14218418945871467103825991766, −0.49964425493072490125387421834, 1.48095251425234716265534826934, 2.60655695855997280987215329255, 3.44749087836661785750565132887, 4.13466273104950444861674502415, 5.06214755876157013257760103599, 6.46480980624158757859443512782, 6.77548446262113259592458978357, 7.57070339968828834644997053076, 8.745394414240392457038587300619, 9.259458664693870515968605546480

Graph of the ZZ-function along the critical line