↑

Properties

Label 2-51-51.47-c2-0-2
Degree $2$
Conductor $51$
Sign $-0.494 - 0.869i$
Analytic cond. $1.38964$
Root an. cond. $1.17883$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 1.28·2-s + (0.662 + 2.92i)3-s − 2.35·4-s + (−2.50 + 2.50i)5-s + (−0.851 − 3.75i)6-s + (−3.77 + 3.77i)7-s + 8.15·8-s + (−8.12 + 3.87i)9-s + (3.22 − 3.22i)10-s + (8.31 + 8.31i)11-s + (−1.55 − 6.87i)12-s + 3.14·13-s + (4.84 − 4.84i)14-s + (−8.99 − 5.67i)15-s − 1.06·16-s + (9.78 − 13.9i)17-s + ⋯
L(s)  = 1  − 0.642·2-s + (0.220 + 0.975i)3-s − 0.587·4-s + (−0.501 + 0.501i)5-s + (−0.141 − 0.626i)6-s + (−0.539 + 0.539i)7-s + 1.01·8-s + (−0.902 + 0.430i)9-s + (0.322 − 0.322i)10-s + (0.755 + 0.755i)11-s + (−0.129 − 0.573i)12-s + 0.241·13-s + (0.346 − 0.346i)14-s + (−0.599 − 0.378i)15-s − 0.0668·16-s + (0.575 − 0.817i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 51 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.494 - 0.869i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.494 - 0.869i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(51\)    =    \(3 \cdot 17\)
Sign: $-0.494 - 0.869i$
Analytic conductor: \(1.38964\)
Root analytic conductor: \(1.17883\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{51} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 51,\ (\ :1),\ -0.494 - 0.869i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.316133 + 0.543678i\)
\(L(\frac12)\) \(\approx\) \(0.316133 + 0.543678i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.662 - 2.92i)T \)
17 \( 1 + (-9.78 + 13.9i)T \)
good2 \( 1 + 1.28T + 4T^{2} \)
5 \( 1 + (2.50 - 2.50i)T - 25iT^{2} \)
7 \( 1 + (3.77 - 3.77i)T - 49iT^{2} \)
11 \( 1 + (-8.31 - 8.31i)T + 121iT^{2} \)
13 \( 1 - 3.14T + 169T^{2} \)
19 \( 1 + 12.4iT - 361T^{2} \)
23 \( 1 + (-12.7 - 12.7i)T + 529iT^{2} \)
29 \( 1 + (27.7 - 27.7i)T - 841iT^{2} \)
31 \( 1 + (-38.3 - 38.3i)T + 961iT^{2} \)
37 \( 1 + (-9.59 - 9.59i)T + 1.36e3iT^{2} \)
41 \( 1 + (19.4 + 19.4i)T + 1.68e3iT^{2} \)
43 \( 1 + 76.3iT - 1.84e3T^{2} \)
47 \( 1 - 4.62iT - 2.20e3T^{2} \)
53 \( 1 - 11.0T + 2.80e3T^{2} \)
59 \( 1 - 90.6T + 3.48e3T^{2} \)
61 \( 1 + (-0.602 + 0.602i)T - 3.72e3iT^{2} \)
67 \( 1 + 122.T + 4.48e3T^{2} \)
71 \( 1 + (-64.8 + 64.8i)T - 5.04e3iT^{2} \)
73 \( 1 + (0.537 + 0.537i)T + 5.32e3iT^{2} \)
79 \( 1 + (-90.5 + 90.5i)T - 6.24e3iT^{2} \)
83 \( 1 - 24.5T + 6.88e3T^{2} \)
89 \( 1 - 36.2iT - 7.92e3T^{2} \)
97 \( 1 + (-8.31 - 8.31i)T + 9.40e3iT^{2} \)
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

−15.64641992976774153272953379237, −14.74969362736491812965930519273, −13.60080211513504529818043674194, −11.95455420991551629670654057781, −10.64936002961057596941675632894, −9.514448831419353107744942273907, −8.834485867845943833491685482853, −7.22772733575975257859312413460, −5.04921929788111506749914362293, −3.47155967695374933627002290862, 0.841413746315322516417052789639, 3.91525289456957796707245914097, 6.20903439451484117220050979291, 7.83740142875856308314137000043, 8.570896360307536154446268474960, 9.873134914960384281224781718032, 11.48808972604953722755427358405, 12.78538166681966657643328965175, 13.56632633844040243550505312673, 14.67091522550410245247835089835

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