↑

Properties

Label 2-882-9.7-c1-0-38
Degree $2$
Conductor $882$
Sign $-0.939 + 0.342i$
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 + (1.5 − 0.866i)3-s + (−0.499 − 0.866i)4-s + (−1.5 − 2.59i)5-s − 1.73i·6-s − 0.999·8-s + (1.5 − 2.59i)9-s − 3·10-s + (1.5 − 2.59i)11-s + (−1.49 − 0.866i)12-s + (0.5 + 0.866i)13-s + (−4.5 − 2.59i)15-s + (−0.5 + 0.866i)16-s + 3·17-s + (−1.5 − 2.59i)18-s − 7·19-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (0.866 − 0.499i)3-s + (−0.249 − 0.433i)4-s + (−0.670 − 1.16i)5-s − 0.707i·6-s − 0.353·8-s + (0.5 − 0.866i)9-s − 0.948·10-s + (0.452 − 0.783i)11-s + (−0.433 − 0.250i)12-s + (0.138 + 0.240i)13-s + (−1.16 − 0.670i)15-s + (−0.125 + 0.216i)16-s + 0.727·17-s + (−0.353 − 0.612i)18-s − 1.60·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.939 + 0.342i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.939 + 0.342i$
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: \((2,\ 882,\ (\ :1/2),\ -0.939 + 0.342i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.357672 - 2.02846i\)
\(L(\frac12)\) \(\approx\) \(0.357672 - 2.02846i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.5 + 0.866i)T \)
7 \( 1 \)
good5 \( 1 + (1.5 + 2.59i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.5 + 2.59i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.5 - 0.866i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 - 3T + 17T^{2} \)
19 \( 1 + 7T + 19T^{2} \)
23 \( 1 + (-4.5 - 7.79i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.5 - 2.59i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (4 + 6.92i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + T + 37T^{2} \)
41 \( 1 + (1.5 + 2.59i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.5 + 0.866i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 3T + 53T^{2} \)
59 \( 1 + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1 - 1.73i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2 - 3.46i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 12T + 71T^{2} \)
73 \( 1 - 11T + 73T^{2} \)
79 \( 1 + (-8 + 13.8i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-4.5 + 7.79i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 3T + 89T^{2} \)
97 \( 1 + (-0.5 + 0.866i)T + (-48.5 - 84.0i)T^{2} \)
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

−9.454079034914210821455610818545, −8.964466080400736325279601625811, −8.268702464061574922066874429894, −7.41951564053884393473794717973, −6.22952331009244406462469637550, −5.14976388106831883345265601515, −3.98410670339024695519271857319, −3.46399557424574506418907330839, −1.95291306326289666601030184946, −0.825068631010804997133027551777, 2.34224608711925779337650361448, 3.36530984004748921913724485055, 4.11881079489351645301307415691, 5.03261473919642270060484418724, 6.54778049645493058687722789851, 7.02350813203831389192549271919, 7.971683925968841737321057256028, 8.608822680785534296988298938067, 9.598992640012087729700307850327, 10.56724178981074658651476480518

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