↑

Properties

Label 2-312-104.77-c3-0-24
Degree 22
Conductor 312312
Sign −0.765−0.644i-0.765 - 0.644i
Analytic cond. 18.408518.4085
Root an. cond. 4.290524.29052
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  + (2.30 + 1.63i)2-s − 3i·3-s + (2.62 + 7.55i)4-s + 3.15·5-s + (4.91 − 6.91i)6-s + 31.8i·7-s + (−6.33 + 21.7i)8-s − 9·9-s + (7.28 + 5.17i)10-s − 30.3·11-s + (22.6 − 7.87i)12-s + (−42.8 − 18.9i)13-s + (−52.1 + 73.3i)14-s − 9.47i·15-s + (−50.2 + 39.6i)16-s + 33.4·17-s + ⋯
L(s)  = 1  + (0.814 + 0.579i)2-s − 0.577i·3-s + (0.328 + 0.944i)4-s + 0.282·5-s + (0.334 − 0.470i)6-s + 1.71i·7-s + (−0.280 + 0.959i)8-s − 0.333·9-s + (0.230 + 0.163i)10-s − 0.831·11-s + (0.545 − 0.189i)12-s + (−0.914 − 0.403i)13-s + (−0.995 + 1.40i)14-s − 0.163i·15-s + (−0.784 + 0.620i)16-s + 0.477·17-s + ⋯

Functional equation

Λ(s)=(312s/2 ΓC(s) L(s)=((−0.765−0.644i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.765 - 0.644i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(312s/2 ΓC(s+3/2) L(s)=((−0.765−0.644i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.765 - 0.644i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 312312    =    23⋅3⋅132^{3} \cdot 3 \cdot 13
Sign: −0.765−0.644i-0.765 - 0.644i
Analytic conductor: 18.408518.4085
Root analytic conductor: 4.290524.29052
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ312(181,⋅)\chi_{312} (181, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 312, ( :3/2), −0.765−0.644i)(2,\ 312,\ (\ :3/2),\ -0.765 - 0.644i)

Particular Values

L(2)L(2) ≈\approx 2.1813638922.181363892
L(12)L(\frac12) ≈\approx 2.1813638922.181363892
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+(−2.30−1.63i)T 1 + (-2.30 - 1.63i)T
3 1+3iT 1 + 3iT
13 1+(42.8+18.9i)T 1 + (42.8 + 18.9i)T
good5 1−3.15T+125T2 1 - 3.15T + 125T^{2}
7 1−31.8iT−343T2 1 - 31.8iT - 343T^{2}
11 1+30.3T+1.33e3T2 1 + 30.3T + 1.33e3T^{2}
17 1−33.4T+4.91e3T2 1 - 33.4T + 4.91e3T^{2}
19 1+32.4T+6.85e3T2 1 + 32.4T + 6.85e3T^{2}
23 1−92.1T+1.21e4T2 1 - 92.1T + 1.21e4T^{2}
29 1+11.6iT−2.43e4T2 1 + 11.6iT - 2.43e4T^{2}
31 1−328.iT−2.97e4T2 1 - 328. iT - 2.97e4T^{2}
37 1−271.T+5.06e4T2 1 - 271.T + 5.06e4T^{2}
41 1−153.iT−6.89e4T2 1 - 153. iT - 6.89e4T^{2}
43 1+26.1iT−7.95e4T2 1 + 26.1iT - 7.95e4T^{2}
47 1+72.2iT−1.03e5T2 1 + 72.2iT - 1.03e5T^{2}
53 1−666.iT−1.48e5T2 1 - 666. iT - 1.48e5T^{2}
59 1−512.T+2.05e5T2 1 - 512.T + 2.05e5T^{2}
61 1+527.iT−2.26e5T2 1 + 527. iT - 2.26e5T^{2}
67 1−863.T+3.00e5T2 1 - 863.T + 3.00e5T^{2}
71 1+810.iT−3.57e5T2 1 + 810. iT - 3.57e5T^{2}
73 1−157.iT−3.89e5T2 1 - 157. iT - 3.89e5T^{2}
79 1−796.T+4.93e5T2 1 - 796.T + 4.93e5T^{2}
83 1−69.5T+5.71e5T2 1 - 69.5T + 5.71e5T^{2}
89 1−1.63e3iT−7.04e5T2 1 - 1.63e3iT - 7.04e5T^{2}
97 1+904.iT−9.12e5T2 1 + 904. iT - 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

−12.11581141192764893562146690843, −10.98395205473348751355055468120, −9.520439086257370532203801844613, −8.460155341715717213857930524257, −7.71959232544806552857610537059, −6.54823197114586738364337846327, −5.58185220359644124936355499772, −5.01027762790665390293750758906, −3.01453453683715713928464294288, −2.23896178067666223228653502375, 0.56751644935740964141343668151, 2.35078844318921428069062080928, 3.75394986958053568543166806119, 4.53603541389453771283699825542, 5.56409285141477231099783790725, 6.86326702476700258150113196485, 7.81884975615745781726973232631, 9.635147580454000534455052244731, 10.03737385537914793952315703128, 10.89990860290838448240169511491

Graph of the ZZ-function along the critical line