↑

Properties

Label 2-51-51.50-c4-0-14
Degree $2$
Conductor $51$
Sign $0.239 + 0.970i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 51 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.239 + 0.970i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.239 + 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(51\)    =    \(3 \cdot 17\)
Sign: $0.239 + 0.970i$
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: \((2,\ 51,\ (\ :2),\ 0.239 + 0.970i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.83724 - 1.43955i\)
\(L(\frac12)\) \(\approx\) \(1.83724 - 1.43955i\)
\(L(3)\) 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

−14.40831519672928346535141532210, −12.86350690131023732282585178148, −12.57163689960264672400935761819, −11.13211610687893879744554239814, −9.661474356904405924635177727605, −8.685289823547330837189231405165, −6.66089636375318072086582613905, −5.81313337775177441295664839260, −2.48824260897433841201703117149, −1.96970331255723460844894091607, 2.34912742108660024606512917824, 4.61651158061171551897404842301, 6.13324319266543052193340941743, 7.39833333542932859676937581632, 9.093512674854452392456070355574, 10.27612323151716740038598608350, 10.94598372704315470414556061132, 13.23567750958922079497556036025, 14.13768610914980252748609122143, 14.84737043858097954447627036469

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