↑

Properties

Label 2-51-51.50-c4-0-2
Degree $2$
Conductor $51$
Sign $-0.866 - 0.498i$
Analytic cond. $5.27186$
Root an. cond. $2.29605$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2.64i·2-s + (−3.53 − 8.27i)3-s + 9.00·4-s − 41.0·5-s + (21.8 − 9.35i)6-s + 79.2i·7-s + 66.1i·8-s + (−56.0 + 58.5i)9-s − 108. i·10-s − 10.0·11-s + (−31.8 − 74.4i)12-s − 126.·13-s − 209.·14-s + (145. + 339. i)15-s − 30.9·16-s + (230. − 173. i)17-s + ⋯
L(s)  = 1  + 0.661i·2-s + (−0.392 − 0.919i)3-s + 0.562·4-s − 1.64·5-s + (0.608 − 0.259i)6-s + 1.61i·7-s + 1.03i·8-s + (−0.691 + 0.722i)9-s − 1.08i·10-s − 0.0832·11-s + (−0.220 − 0.517i)12-s − 0.749·13-s − 1.06·14-s + (0.645 + 1.51i)15-s − 0.120·16-s + (0.798 − 0.601i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 51 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.866 - 0.498i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.866 - 0.498i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(51\)    =    \(3 \cdot 17\)
Sign: $-0.866 - 0.498i$
Analytic conductor: \(5.27186\)
Root analytic conductor: \(2.29605\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{51} (50, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 51,\ (\ :2),\ -0.866 - 0.498i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.156724 + 0.586760i\)
\(L(\frac12)\) \(\approx\) \(0.156724 + 0.586760i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (3.53 + 8.27i)T \)
17 \( 1 + (-230. + 173. i)T \)
good2 \( 1 - 2.64iT - 16T^{2} \)
5 \( 1 + 41.0T + 625T^{2} \)
7 \( 1 - 79.2iT - 2.40e3T^{2} \)
11 \( 1 + 10.0T + 1.46e4T^{2} \)
13 \( 1 + 126.T + 2.85e4T^{2} \)
19 \( 1 + 553.T + 1.30e5T^{2} \)
23 \( 1 + 152.T + 2.79e5T^{2} \)
29 \( 1 - 301.T + 7.07e5T^{2} \)
31 \( 1 - 342. iT - 9.23e5T^{2} \)
37 \( 1 + 133. iT - 1.87e6T^{2} \)
41 \( 1 + 2.02e3T + 2.82e6T^{2} \)
43 \( 1 - 1.49e3T + 3.41e6T^{2} \)
47 \( 1 - 451. iT - 4.87e6T^{2} \)
53 \( 1 - 2.82e3iT - 7.89e6T^{2} \)
59 \( 1 - 1.24e3iT - 1.21e7T^{2} \)
61 \( 1 + 2.18e3iT - 1.38e7T^{2} \)
67 \( 1 - 5.08e3T + 2.01e7T^{2} \)
71 \( 1 - 4.12e3T + 2.54e7T^{2} \)
73 \( 1 - 1.69e3iT - 2.83e7T^{2} \)
79 \( 1 - 1.10e4iT - 3.89e7T^{2} \)
83 \( 1 + 1.19e4iT - 4.74e7T^{2} \)
89 \( 1 - 1.36e4iT - 6.27e7T^{2} \)
97 \( 1 - 5.21e3iT - 8.85e7T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−15.33641545951758291485840457892, −14.52220380380375932430906754779, −12.33604240143802544025256810519, −12.08554185308942553768839968182, −11.05021390470931292814478483066, −8.520208328388900894671052557778, −7.75182080938957812808759525962, −6.59696191991323926186529951135, −5.23631265479648478315000201032, −2.55020241492113276374934490023, 0.36356770968680621380221856203, 3.55301197500954418434980589820, 4.33887995703699860971527737146, 6.81409953814302994250422727874, 8.022718814214034373068750363681, 10.12102344099443243482423828048, 10.76225232810275751013628870324, 11.67468693415480711398732394763, 12.64704439131479912355601625599, 14.61042499382134324011856874651

Graph of the $Z$-function along the critical line