↑

Properties

Label 2-96-8.5-c3-0-2
Degree 22
Conductor 9696
Sign 0.427−0.903i0.427 - 0.903i
Analytic cond. 5.664185.66418
Root an. cond. 2.379952.37995
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 3i·3-s − 0.612i·5-s + 22.7·7-s − 9·9-s + 60.2i·11-s + 52.9i·13-s + 1.83·15-s + 47.1·17-s − 29.1i·19-s + 68.2i·21-s − 109.·23-s + 124.·25-s − 27i·27-s − 10.4i·29-s + 220.·31-s + ⋯
L(s)  = 1  + 0.577i·3-s − 0.0547i·5-s + 1.22·7-s − 0.333·9-s + 1.65i·11-s + 1.12i·13-s + 0.0316·15-s + 0.672·17-s − 0.352i·19-s + 0.709i·21-s − 0.992·23-s + 0.996·25-s − 0.192i·27-s − 0.0667i·29-s + 1.27·31-s + ⋯

Functional equation

Λ(s)=(96s/2 ΓC(s) L(s)=((0.427−0.903i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 96 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.427 - 0.903i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(96s/2 ΓC(s+3/2) L(s)=((0.427−0.903i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 96 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.427 - 0.903i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 9696    =    25⋅32^{5} \cdot 3
Sign: 0.427−0.903i0.427 - 0.903i
Analytic conductor: 5.664185.66418
Root analytic conductor: 2.379952.37995
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ96(49,⋅)\chi_{96} (49, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 96, ( :3/2), 0.427−0.903i)(2,\ 96,\ (\ :3/2),\ 0.427 - 0.903i)

Particular Values

L(2)L(2) ≈\approx 1.40024+0.886509i1.40024 + 0.886509i
L(12)L(\frac12) ≈\approx 1.40024+0.886509i1.40024 + 0.886509i
L(52)L(\frac{5}{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 1−3iT 1 - 3iT
good5 1+0.612iT−125T2 1 + 0.612iT - 125T^{2}
7 1−22.7T+343T2 1 - 22.7T + 343T^{2}
11 1−60.2iT−1.33e3T2 1 - 60.2iT - 1.33e3T^{2}
13 1−52.9iT−2.19e3T2 1 - 52.9iT - 2.19e3T^{2}
17 1−47.1T+4.91e3T2 1 - 47.1T + 4.91e3T^{2}
19 1+29.1iT−6.85e3T2 1 + 29.1iT - 6.85e3T^{2}
23 1+109.T+1.21e4T2 1 + 109.T + 1.21e4T^{2}
29 1+10.4iT−2.43e4T2 1 + 10.4iT - 2.43e4T^{2}
31 1−220.T+2.97e4T2 1 - 220.T + 2.97e4T^{2}
37 1+408.iT−5.06e4T2 1 + 408. iT - 5.06e4T^{2}
41 1+360.T+6.89e4T2 1 + 360.T + 6.89e4T^{2}
43 1+236.iT−7.95e4T2 1 + 236. iT - 7.95e4T^{2}
47 1+129.T+1.03e5T2 1 + 129.T + 1.03e5T^{2}
53 1+117.iT−1.48e5T2 1 + 117. iT - 1.48e5T^{2}
59 1+262.iT−2.05e5T2 1 + 262. iT - 2.05e5T^{2}
61 1−273.iT−2.26e5T2 1 - 273. iT - 2.26e5T^{2}
67 1+89.4iT−3.00e5T2 1 + 89.4iT - 3.00e5T^{2}
71 1−350.T+3.57e5T2 1 - 350.T + 3.57e5T^{2}
73 1−532.T+3.89e5T2 1 - 532.T + 3.89e5T^{2}
79 1−166.T+4.93e5T2 1 - 166.T + 4.93e5T^{2}
83 1−361.iT−5.71e5T2 1 - 361. iT - 5.71e5T^{2}
89 1−40.3T+7.04e5T2 1 - 40.3T + 7.04e5T^{2}
97 1+614.T+9.12e5T2 1 + 614.T + 9.12e5T^{2}
show more
show less
   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

−13.99196963554436405536025241066, −12.37128819844717534980187252489, −11.57905096378395016030190116622, −10.39166112859402680759411472612, −9.393208089701774574074209121972, −8.163604944758821765782660197407, −6.92620319782875688718453191580, −5.10358281834891909894618662917, −4.22645757138495605512171968302, −1.96172237674734729345538275186, 1.09999676689116059091841406697, 3.12210026545588998535672609522, 5.12778777022101366021161096721, 6.28294529778078933220065088040, 8.037388212965681409055556069259, 8.333847521904450220817807060741, 10.24235431740687672921695475086, 11.27171268945163775357138239647, 12.13128679009365501666463664687, 13.44311101427324408421837291811

Graph of the ZZ-function along the critical line