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Properties

Label 2-882-7.2-c1-0-10
Degree $2$
Conductor $882$
Sign $-0.0725 + 0.997i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0725 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0725 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.0725 + 0.997i$
Analytic conductor: \(7.04280\)
Root analytic conductor: \(2.65382\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{882} (667, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 882,\ (\ :1/2),\ -0.0725 + 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.36256 - 1.46529i\)
\(L(\frac12)\) \(\approx\) \(1.36256 - 1.46529i\)
\(L(\frac{3}{2})\) 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

−9.880663487635325915788117309287, −9.102331844859187705086039445426, −8.668118434840860527742860830945, −7.10195216315226775515316139810, −6.53303832332096410301683178964, −5.08095414495171742943271294648, −4.72190497129070411274419810112, −3.44051663652896944893109986483, −2.25057248695087158925981099390, −0.968702465333726216086743931593, 1.57538423239937573055979176463, 3.35751949347075180898923514539, 3.84094846432175063535720605690, 5.34915736567075376994640027468, 6.15922629835446783543826966800, 6.62218823392098014037216335273, 7.83444342483286895262022980837, 8.578712605614805880843543811557, 9.306238181175000976577847637620, 10.50865238062249577287737902029

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