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Properties

Label 2-882-63.58-c1-0-4
Degree $2$
Conductor $882$
Sign $0.512 - 0.858i$
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 + (0.5 − 1.65i)3-s + 4-s + (0.686 + 1.18i)5-s + (−0.5 + 1.65i)6-s − 8-s + (−2.5 − 1.65i)9-s + (−0.686 − 1.18i)10-s + (−2.18 + 3.78i)11-s + (0.5 − 1.65i)12-s + (−1 + 1.73i)13-s + (2.31 − 0.543i)15-s + 16-s + (2.18 + 3.78i)17-s + (2.5 + 1.65i)18-s + (−2.5 + 4.33i)19-s + ⋯
L(s)  = 1  − 0.707·2-s + (0.288 − 0.957i)3-s + 0.5·4-s + (0.306 + 0.531i)5-s + (−0.204 + 0.677i)6-s − 0.353·8-s + (−0.833 − 0.552i)9-s + (−0.216 − 0.375i)10-s + (−0.659 + 1.14i)11-s + (0.144 − 0.478i)12-s + (−0.277 + 0.480i)13-s + (0.597 − 0.140i)15-s + 0.250·16-s + (0.530 + 0.918i)17-s + (0.589 + 0.390i)18-s + (−0.573 + 0.993i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.512 - 0.858i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.512 - 0.858i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.512 - 0.858i$
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: \((2,\ 882,\ (\ :1/2),\ 0.512 - 0.858i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.829635 + 0.471111i\)
\(L(\frac12)\) \(\approx\) \(0.829635 + 0.471111i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 + 1.65i)T \)
7 \( 1 \)
good5 \( 1 + (-0.686 - 1.18i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (2.18 - 3.78i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (1 - 1.73i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2.18 - 3.78i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.5 - 4.33i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.68 - 6.38i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.37 + 2.37i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 2T + 31T^{2} \)
37 \( 1 + (1 - 1.73i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (5.18 - 8.98i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.55 + 7.89i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + (1.37 + 2.37i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 7.11T + 59T^{2} \)
61 \( 1 + 14.1T + 61T^{2} \)
67 \( 1 - 15.1T + 67T^{2} \)
71 \( 1 - 10.1T + 71T^{2} \)
73 \( 1 + (-2.55 - 4.43i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 - 12.1T + 79T^{2} \)
83 \( 1 + (-2.74 - 4.75i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (1.62 - 2.81i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (4.55 + 7.89i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.08498847338183296611826070779, −9.548938165652590128923197738094, −8.353121379114997056939923588990, −7.82558811654136268725759205584, −6.94113952076478391115287803848, −6.34965658173622514145795234122, −5.23492794995606086474644387144, −3.58716830092482537799373396611, −2.37287115006211072697219471491, −1.57554304970466362172548748094, 0.54591743175343111291134740029, 2.53131245205573829950108739792, 3.32663057106721408303186125694, 4.88867815302078554668618865899, 5.37362729261151268202039248911, 6.60574952439824204582245878961, 7.78964098085426331248647046811, 8.610687756488483824718444526070, 9.073932146533817893978555988165, 9.904079564750673309936203902077

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