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Properties

Label 2-882-63.4-c1-0-14
Degree $2$
Conductor $882$
Sign $0.417 - 0.908i$
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 + (−0.619 + 1.61i)3-s + (−0.499 − 0.866i)4-s − 1.76·5-s + (−1.09 − 1.34i)6-s + 0.999·8-s + (−2.23 − 2.00i)9-s + (0.880 − 1.52i)10-s + 6.12·11-s + (1.71 − 0.272i)12-s + (0.380 − 0.658i)13-s + (1.09 − 2.84i)15-s + (−0.5 + 0.866i)16-s + (3.42 − 5.92i)17-s + (2.85 − 0.931i)18-s + (−0.971 − 1.68i)19-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.357 + 0.933i)3-s + (−0.249 − 0.433i)4-s − 0.787·5-s + (−0.445 − 0.549i)6-s + 0.353·8-s + (−0.744 − 0.668i)9-s + (0.278 − 0.482i)10-s + 1.84·11-s + (0.493 − 0.0785i)12-s + (0.105 − 0.182i)13-s + (0.281 − 0.735i)15-s + (−0.125 + 0.216i)16-s + (0.829 − 1.43i)17-s + (0.672 − 0.219i)18-s + (−0.222 − 0.385i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.417 - 0.908i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.417 - 0.908i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.417 - 0.908i$
Analytic conductor: \(7.04280\)
Root analytic conductor: \(2.65382\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{882} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 882,\ (\ :1/2),\ 0.417 - 0.908i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.854525 + 0.548003i\)
\(L(\frac12)\) \(\approx\) \(0.854525 + 0.548003i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.619 - 1.61i)T \)
7 \( 1 \)
good5 \( 1 + 1.76T + 5T^{2} \)
11 \( 1 - 6.12T + 11T^{2} \)
13 \( 1 + (-0.380 + 0.658i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-3.42 + 5.92i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.971 + 1.68i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 0.421T + 23T^{2} \)
29 \( 1 + (-0.732 - 1.26i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3.85 - 6.67i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.44 - 2.49i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-3.47 + 6.01i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4.33 - 7.49i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-0.830 + 1.43i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.112 - 0.195i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-0.993 - 1.72i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (5.17 - 8.96i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.39 + 5.87i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 10.7T + 71T^{2} \)
73 \( 1 + (0.153 - 0.265i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-6.72 + 11.6i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-1.56 - 2.70i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (1.30 + 2.25i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-1.81 - 3.14i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.11816629835095632867682156541, −9.300226592069166644846459702098, −8.834820081498663541697626940087, −7.74880046678592397260791823992, −6.83357082895737276285331335283, −6.02873743838603310345288814761, −4.92866719286475898525443300046, −4.15495510407590828095756576275, −3.20201075350889198595479878384, −0.860120525913748664708409208595, 0.948135128633865970709958598002, 2.01170928621604340445318628659, 3.59743946881110105349437691509, 4.25815264067419351492449065100, 5.89467695074466946112461881623, 6.55211242284378579004622306134, 7.66939033615297648355464918197, 8.184936931679901370750976282658, 9.087779351970945710306903844760, 10.05250257774211129511283110208

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