↑

Properties

Label 2-798-1.1-c3-0-1
Degree 22
Conductor 798798
Sign 11
Analytic cond. 47.083547.0835
Root an. cond. 6.861746.86174
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s − 3·3-s + 4·4-s − 2.35·5-s + 6·6-s − 7·7-s − 8·8-s + 9·9-s + 4.70·10-s − 65.1·11-s − 12·12-s − 31.4·13-s + 14·14-s + 7.05·15-s + 16·16-s − 43.0·17-s − 18·18-s − 19·19-s − 9.40·20-s + 21·21-s + 130.·22-s + 92.8·23-s + 24·24-s − 119.·25-s + 62.9·26-s − 27·27-s − 28·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.210·5-s + 0.408·6-s − 0.377·7-s − 0.353·8-s + 0.333·9-s + 0.148·10-s − 1.78·11-s − 0.288·12-s − 0.671·13-s + 0.267·14-s + 0.121·15-s + 0.250·16-s − 0.614·17-s − 0.235·18-s − 0.229·19-s − 0.105·20-s + 0.218·21-s + 1.26·22-s + 0.841·23-s + 0.204·24-s − 0.955·25-s + 0.474·26-s − 0.192·27-s − 0.188·28-s + ⋯

Functional equation

Λ(s)=(798s/2 ΓC(s) L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(798s/2 ΓC(s+3/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 798798    =    2⋅3⋅7⋅192 \cdot 3 \cdot 7 \cdot 19
Sign: 11
Analytic conductor: 47.083547.0835
Root analytic conductor: 6.861746.86174
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 798, ( :3/2), 1)(2,\ 798,\ (\ :3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 0.36701779870.3670177987
L(12)L(\frac12) ≈\approx 0.36701779870.3670177987
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+2T 1 + 2T
3 1+3T 1 + 3T
7 1+7T 1 + 7T
19 1+19T 1 + 19T
good5 1+2.35T+125T2 1 + 2.35T + 125T^{2}
11 1+65.1T+1.33e3T2 1 + 65.1T + 1.33e3T^{2}
13 1+31.4T+2.19e3T2 1 + 31.4T + 2.19e3T^{2}
17 1+43.0T+4.91e3T2 1 + 43.0T + 4.91e3T^{2}
23 1−92.8T+1.21e4T2 1 - 92.8T + 1.21e4T^{2}
29 1+95.7T+2.43e4T2 1 + 95.7T + 2.43e4T^{2}
31 1−161.T+2.97e4T2 1 - 161.T + 2.97e4T^{2}
37 1−5.04T+5.06e4T2 1 - 5.04T + 5.06e4T^{2}
41 1+434.T+6.89e4T2 1 + 434.T + 6.89e4T^{2}
43 1−54.7T+7.95e4T2 1 - 54.7T + 7.95e4T^{2}
47 1+220.T+1.03e5T2 1 + 220.T + 1.03e5T^{2}
53 1−175.T+1.48e5T2 1 - 175.T + 1.48e5T^{2}
59 1+383.T+2.05e5T2 1 + 383.T + 2.05e5T^{2}
61 1−431.T+2.26e5T2 1 - 431.T + 2.26e5T^{2}
67 1+989.T+3.00e5T2 1 + 989.T + 3.00e5T^{2}
71 1+456.T+3.57e5T2 1 + 456.T + 3.57e5T^{2}
73 1−92.3T+3.89e5T2 1 - 92.3T + 3.89e5T^{2}
79 1−620.T+4.93e5T2 1 - 620.T + 4.93e5T^{2}
83 1−519.T+5.71e5T2 1 - 519.T + 5.71e5T^{2}
89 1−1.04e3T+7.04e5T2 1 - 1.04e3T + 7.04e5T^{2}
97 1+166.T+9.12e5T2 1 + 166.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

−10.07975305188265469799897379704, −9.095486358935142479205118957160, −8.075587467686266991987323661806, −7.41034129168171869887621587404, −6.52776980428650501183835231446, −5.49714141096704188179023152033, −4.65051666470963939744322995374, −3.13202129984313756749843177126, −2.06926443399384873153478546466, −0.36498004466784405916452090718, 0.36498004466784405916452090718, 2.06926443399384873153478546466, 3.13202129984313756749843177126, 4.65051666470963939744322995374, 5.49714141096704188179023152033, 6.52776980428650501183835231446, 7.41034129168171869887621587404, 8.075587467686266991987323661806, 9.095486358935142479205118957160, 10.07975305188265469799897379704

Graph of the ZZ-function along the critical line