↑

Properties

Label 2-882-63.41-c1-0-14
Degree $2$
Conductor $882$
Sign $0.701 - 0.712i$
Analytic cond. $7.04280$
Root an. cond. $2.65382$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.866 + 0.5i)2-s + (1.59 − 0.681i)3-s + (0.499 − 0.866i)4-s + (−0.714 + 1.23i)5-s + (−1.03 + 1.38i)6-s + 0.999i·8-s + (2.07 − 2.17i)9-s − 1.42i·10-s + (−2.96 + 1.70i)11-s + (0.206 − 1.71i)12-s + (5.48 + 3.16i)13-s + (−0.294 + 2.45i)15-s + (−0.5 − 0.866i)16-s − 2.28·17-s + (−0.708 + 2.91i)18-s − 2.16i·19-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.919 − 0.393i)3-s + (0.249 − 0.433i)4-s + (−0.319 + 0.553i)5-s + (−0.423 + 0.565i)6-s + 0.353i·8-s + (0.690 − 0.723i)9-s − 0.452i·10-s + (−0.892 + 0.515i)11-s + (0.0594 − 0.496i)12-s + (1.52 + 0.878i)13-s + (−0.0760 + 0.634i)15-s + (−0.125 − 0.216i)16-s − 0.553·17-s + (−0.167 + 0.687i)18-s − 0.497i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.701 - 0.712i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.701 - 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.701 - 0.712i$
Analytic conductor: \(7.04280\)
Root analytic conductor: \(2.65382\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{882} (293, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 882,\ (\ :1/2),\ 0.701 - 0.712i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.42434 + 0.596405i\)
\(L(\frac12)\) \(\approx\) \(1.42434 + 0.596405i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 - 0.5i)T \)
3 \( 1 + (-1.59 + 0.681i)T \)
7 \( 1 \)
good5 \( 1 + (0.714 - 1.23i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (2.96 - 1.70i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-5.48 - 3.16i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + 2.28T + 17T^{2} \)
19 \( 1 + 2.16iT - 19T^{2} \)
23 \( 1 + (-6.97 - 4.02i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.298 - 0.172i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-3.76 - 2.17i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + 2.15T + 37T^{2} \)
41 \( 1 + (0.202 - 0.350i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.90 - 5.03i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (2.75 + 4.77i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 9.88iT - 53T^{2} \)
59 \( 1 + (-5.51 + 9.55i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-9.94 + 5.73i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.12 - 3.68i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 3.55iT - 71T^{2} \)
73 \( 1 - 0.232iT - 73T^{2} \)
79 \( 1 + (7.28 + 12.6i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.811 + 1.40i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 4.05T + 89T^{2} \)
97 \( 1 + (-9.18 + 5.30i)T + (48.5 - 84.0i)T^{2} \)
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

−10.08731546398990067921788562140, −9.090381097387914821948537628842, −8.660201998585122885566425733189, −7.66884948294345094664576691074, −7.01765453944542912593552962324, −6.37709989907637661183344804455, −4.94131973614177076907902370178, −3.65919003968874066784519450425, −2.64633661360778512235650482947, −1.37925347969871100902780228075, 0.942276407603216963042631486671, 2.53264813417342685814344712667, 3.41611790471131309462250585582, 4.41332599357883119025763259328, 5.54993361167998594101170745171, 6.86005304704800607968829178123, 8.019009843080315113505161066080, 8.464480588020801993616141751452, 8.916138744031926907253028481786, 10.09893345862168230760005229979

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