Properties

Label 2-882-7.4-c1-0-2
Degree $2$
Conductor $882$
Sign $0.386 - 0.922i$
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 + (−2 − 3.46i)5-s + 0.999·8-s + (−1.99 + 3.46i)10-s + (−2 + 3.46i)11-s − 4·13-s + (−0.5 − 0.866i)16-s + (2 + 3.46i)19-s + 3.99·20-s + 3.99·22-s + (−5.49 + 9.52i)25-s + (2 + 3.46i)26-s − 2·29-s + (4 − 6.92i)31-s + (−0.499 + 0.866i)32-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + (−0.894 − 1.54i)5-s + 0.353·8-s + (−0.632 + 1.09i)10-s + (−0.603 + 1.04i)11-s − 1.10·13-s + (−0.125 − 0.216i)16-s + (0.458 + 0.794i)19-s + 0.894·20-s + 0.852·22-s + (−1.09 + 1.90i)25-s + (0.392 + 0.679i)26-s − 0.371·29-s + (0.718 − 1.24i)31-s + (−0.0883 + 0.153i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.386 - 0.922i$
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: \((2,\ 882,\ (\ :1/2),\ 0.386 - 0.922i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.253791 + 0.168817i\)
\(L(\frac12)\) \(\approx\) \(0.253791 + 0.168817i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2 + 3.46i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (2 - 3.46i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 4T + 13T^{2} \)
17 \( 1 + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2 - 3.46i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 2T + 29T^{2} \)
31 \( 1 + (-4 + 6.92i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3 - 5.19i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 - 4T + 43T^{2} \)
47 \( 1 + (-4 - 6.92i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (5 - 8.66i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (2 - 3.46i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2 + 3.46i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2 - 3.46i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 8T + 71T^{2} \)
73 \( 1 + (8 - 13.8i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-4 - 6.92i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 12T + 83T^{2} \)
89 \( 1 + (4 + 6.92i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 8T + 97T^{2} \)
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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

−10.01765387411818538205262660060, −9.589335222697759326356984041191, −8.671694418567214377795220923885, −7.76095457887372807068681851025, −7.45534075042396885870294403366, −5.66525890447160139716469522572, −4.63846684489140837676416592443, −4.20615279414138285440535719259, −2.68026711556100922913687148197, −1.30570357701900458895636339923, 0.17513715502348679575213055109, 2.59538186416765274330288557033, 3.41707655795357253114133088661, 4.72054042168802117219810592524, 5.82252664519679312929149671366, 6.80569797625361400590801836675, 7.39144694042487636358131614868, 8.035946922846158562595707519347, 9.029143982657695932303551230245, 10.10549202492995439153931728270

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