↑

Properties

Label 2-882-9.7-c1-0-0
Degree $2$
Conductor $882$
Sign $-0.989 + 0.142i$
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  + (−0.5 + 0.866i)2-s + (−0.707 − 1.58i)3-s + (−0.499 − 0.866i)4-s + (1.01 + 1.75i)5-s + (1.72 + 0.178i)6-s + 0.999·8-s + (−2.00 + 2.23i)9-s − 2.03·10-s + (−2 + 3.46i)11-s + (−1.01 + 1.40i)12-s + (−2.12 − 3.67i)13-s + (2.06 − 2.85i)15-s + (−0.5 + 0.866i)16-s − 1.41·17-s + (−0.936 − 2.85i)18-s + 0.796·19-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.408 − 0.912i)3-s + (−0.249 − 0.433i)4-s + (0.454 + 0.786i)5-s + (0.703 + 0.0727i)6-s + 0.353·8-s + (−0.666 + 0.745i)9-s − 0.642·10-s + (−0.603 + 1.04i)11-s + (−0.293 + 0.404i)12-s + (−0.588 − 1.01i)13-s + (0.532 − 0.735i)15-s + (−0.125 + 0.216i)16-s − 0.342·17-s + (−0.220 − 0.671i)18-s + 0.182·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.989 + 0.142i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.989 + 0.142i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.989 + 0.142i$
Analytic conductor: \(7.04280\)
Root analytic conductor: \(2.65382\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{882} (295, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 882,\ (\ :1/2),\ -0.989 + 0.142i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00672822 - 0.0939822i\)
\(L(\frac12)\) \(\approx\) \(0.00672822 - 0.0939822i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 - 0.866i)T \)
3 \( 1 + (0.707 + 1.58i)T \)
7 \( 1 \)
good5 \( 1 + (-1.01 - 1.75i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (2 - 3.46i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.12 + 3.67i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 1.41T + 17T^{2} \)
19 \( 1 - 0.796T + 19T^{2} \)
23 \( 1 + (3.37 + 5.84i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (4.43 - 7.68i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (2.73 + 4.74i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 9.74T + 37T^{2} \)
41 \( 1 + (-2.82 - 4.89i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (4.43 - 7.68i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-3.44 + 5.96i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 10.6T + 53T^{2} \)
59 \( 1 + (-1.32 - 2.29i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.398 + 0.690i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.436 + 0.756i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 2.12T + 71T^{2} \)
73 \( 1 + 15.3T + 73T^{2} \)
79 \( 1 + (2.93 - 5.08i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-4.15 + 7.19i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 7.07T + 89T^{2} \)
97 \( 1 + (8.92 - 15.4i)T + (-48.5 - 84.0i)T^{2} \)
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

−10.42736585592863845753761960767, −9.918602788132342764777114673789, −8.674368218911709825456797639024, −7.75705827731854750276088931396, −7.17638712848469820413356053797, −6.46356531542142885740771807452, −5.58122483390616320927822002513, −4.75481952891473161771888985589, −2.86014410267444701403920204236, −1.86580650163803274407110774488, 0.05022821053127145607746697280, 1.80140645241199436282320267297, 3.25827116116025203580688435793, 4.22819392347535018427962104358, 5.22718340530239183459986013895, 5.85293793557852939763887049652, 7.23655740981302815937732740300, 8.445863958788668684618499922663, 9.110677914010829038516434881601, 9.654160594191816077966033833537

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