Properties

Label 2-882-7.4-c1-0-13
Degree $2$
Conductor $882$
Sign $0.749 + 0.661i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 + 0.866i)2-s + (−0.499 + 0.866i)4-s + (−0.707 − 1.22i)5-s − 0.999·8-s + (0.707 − 1.22i)10-s + (2 − 3.46i)11-s − 4.24·13-s + (−0.5 − 0.866i)16-s + (3.53 − 6.12i)17-s + (−2.82 − 4.89i)19-s + 1.41·20-s + 3.99·22-s + (4 + 6.92i)23-s + (1.50 − 2.59i)25-s + (−2.12 − 3.67i)26-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (−0.316 − 0.547i)5-s − 0.353·8-s + (0.223 − 0.387i)10-s + (0.603 − 1.04i)11-s − 1.17·13-s + (−0.125 − 0.216i)16-s + (0.857 − 1.48i)17-s + (−0.648 − 1.12i)19-s + 0.316·20-s + 0.852·22-s + (0.834 + 1.44i)23-s + (0.300 − 0.519i)25-s + (−0.416 − 0.720i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.749 + 0.661i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.749 + 0.661i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.749 + 0.661i$
Analytic conductor: \(7.04280\)
Root analytic conductor: \(2.65382\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{882} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 882,\ (\ :1/2),\ 0.749 + 0.661i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.35015 - 0.510496i\)
\(L(\frac12)\) \(\approx\) \(1.35015 - 0.510496i\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 \)
good5 \( 1 + (0.707 + 1.22i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-2 + 3.46i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 4.24T + 13T^{2} \)
17 \( 1 + (-3.53 + 6.12i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.82 + 4.89i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-4 - 6.92i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 2T + 29T^{2} \)
31 \( 1 + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (2 + 3.46i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 9.89T + 41T^{2} \)
43 \( 1 + 4T + 43T^{2} \)
47 \( 1 + (2.82 + 4.89i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2 + 3.46i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-5.65 + 9.79i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.707 - 1.22i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6 + 10.3i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + (7.77 - 13.4i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-8 - 13.8i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 5.65T + 83T^{2} \)
89 \( 1 + (3.53 + 6.12i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 7.07T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.698402288454814678915728451567, −9.154157272965694053195862883740, −8.288404513018746027060612632079, −7.36689535372546371351963268172, −6.73526373731699038662510680285, −5.43717091389875276575135105337, −4.94791757740060135942024572565, −3.78796800478002405974273529316, −2.71468604343279408383827100321, −0.64003733616123595950867641556, 1.60008844771083753488360983081, 2.76337635700734706949798087235, 3.89932083057343143078105657481, 4.64203502502904655967286817085, 5.83691878970931258526636267259, 6.78634092939243797916736960045, 7.62297980968268921404774430808, 8.636829838622846735901098472210, 9.694893583986742279264495710637, 10.32739137098627292792835184908

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