Properties

Label 2-882-63.58-c1-0-31
Degree $2$
Conductor $882$
Sign $0.961 + 0.275i$
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  + 2-s + (1.72 + 0.158i)3-s + 4-s + (−0.724 − 1.25i)5-s + (1.72 + 0.158i)6-s + 8-s + (2.94 + 0.548i)9-s + (−0.724 − 1.25i)10-s + (−1 + 1.73i)11-s + (1.72 + 0.158i)12-s + (2.44 − 4.24i)13-s + (−1.05 − 2.28i)15-s + 16-s + (1 + 1.73i)17-s + (2.94 + 0.548i)18-s + (1.27 − 2.20i)19-s + ⋯
L(s)  = 1  + 0.707·2-s + (0.995 + 0.0917i)3-s + 0.5·4-s + (−0.324 − 0.561i)5-s + (0.704 + 0.0648i)6-s + 0.353·8-s + (0.983 + 0.182i)9-s + (−0.229 − 0.396i)10-s + (−0.301 + 0.522i)11-s + (0.497 + 0.0458i)12-s + (0.679 − 1.17i)13-s + (−0.271 − 0.588i)15-s + 0.250·16-s + (0.242 + 0.420i)17-s + (0.695 + 0.129i)18-s + (0.292 − 0.506i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.961 + 0.275i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.961 + 0.275i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.961 + 0.275i$
Analytic conductor: \(7.04280\)
Root analytic conductor: \(2.65382\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{882} (373, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 882,\ (\ :1/2),\ 0.961 + 0.275i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.28097 - 0.461234i\)
\(L(\frac12)\) \(\approx\) \(3.28097 - 0.461234i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 + (-1.72 - 0.158i)T \)
7 \( 1 \)
good5 \( 1 + (0.724 + 1.25i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (1 - 1.73i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.44 + 4.24i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-1 - 1.73i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.27 + 2.20i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.5 - 0.866i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.44 - 5.97i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 6T + 31T^{2} \)
37 \( 1 + (5.89 - 10.2i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-4.89 + 8.48i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.44 + 5.97i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 9.79T + 47T^{2} \)
53 \( 1 + (-5.44 - 9.43i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 2T + 59T^{2} \)
61 \( 1 + 6.55T + 61T^{2} \)
67 \( 1 + 12.8T + 67T^{2} \)
71 \( 1 - 0.101T + 71T^{2} \)
73 \( 1 + (3.44 + 5.97i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + 1.89T + 79T^{2} \)
83 \( 1 + (-1 - 1.73i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (8.44 - 14.6i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-1.44 - 2.51i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.28519180229230732329990025350, −9.062342318041978882532321046276, −8.393006692999381844047307276151, −7.62124289263485136274145945863, −6.77343874263190645450826834328, −5.44772261471286630358558427001, −4.68383518381932287360637471058, −3.63831447976259961942917756773, −2.86160507481221046413497990912, −1.43098205278338399079499359255, 1.69695934251557328543135873910, 2.94348281852844384207722159660, 3.65594481165534648506263846435, 4.57123502161400594395656962633, 5.88791524679931816619463832628, 6.83262111191519523933700334965, 7.52987215703642594901647504184, 8.403853634853568710424606175587, 9.282732002872879395957759001126, 10.18439378966835514890348239977

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