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Properties

Label 2-882-441.16-c1-0-31
Degree $2$
Conductor $882$
Sign $0.635 + 0.771i$
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  + (0.988 + 0.149i)2-s + (−0.912 + 1.47i)3-s + (0.955 + 0.294i)4-s + (−3.35 − 1.61i)5-s + (−1.12 + 1.31i)6-s + (−1.13 + 2.38i)7-s + (0.900 + 0.433i)8-s + (−1.33 − 2.68i)9-s + (−3.07 − 2.09i)10-s + (2.82 − 3.54i)11-s + (−1.30 + 1.13i)12-s + (−2.57 − 0.387i)13-s + (−1.48 + 2.19i)14-s + (5.44 − 3.46i)15-s + (0.826 + 0.563i)16-s + (2.96 − 0.913i)17-s + ⋯
L(s)  = 1  + (0.699 + 0.105i)2-s + (−0.527 + 0.849i)3-s + (0.477 + 0.147i)4-s + (−1.50 − 0.722i)5-s + (−0.458 + 0.538i)6-s + (−0.430 + 0.902i)7-s + (0.318 + 0.153i)8-s + (−0.444 − 0.895i)9-s + (−0.973 − 0.663i)10-s + (0.851 − 1.06i)11-s + (−0.377 + 0.328i)12-s + (−0.712 − 0.107i)13-s + (−0.396 + 0.585i)14-s + (1.40 − 0.894i)15-s + (0.206 + 0.140i)16-s + (0.718 − 0.221i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.635 + 0.771i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.635 + 0.771i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.635 + 0.771i$
Analytic conductor: \(7.04280\)
Root analytic conductor: \(2.65382\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{882} (457, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 882,\ (\ :1/2),\ 0.635 + 0.771i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.995922 - 0.469884i\)
\(L(\frac12)\) \(\approx\) \(0.995922 - 0.469884i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.988 - 0.149i)T \)
3 \( 1 + (0.912 - 1.47i)T \)
7 \( 1 + (1.13 - 2.38i)T \)
good5 \( 1 + (3.35 + 1.61i)T + (3.11 + 3.90i)T^{2} \)
11 \( 1 + (-2.82 + 3.54i)T + (-2.44 - 10.7i)T^{2} \)
13 \( 1 + (2.57 + 0.387i)T + (12.4 + 3.83i)T^{2} \)
17 \( 1 + (-2.96 + 0.913i)T + (14.0 - 9.57i)T^{2} \)
19 \( 1 + (-3.17 + 5.49i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.820 + 3.59i)T + (-20.7 + 9.97i)T^{2} \)
29 \( 1 + (-4.69 - 4.36i)T + (2.16 + 28.9i)T^{2} \)
31 \( 1 + (0.909 - 1.57i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.98 + 2.76i)T + (2.76 + 36.8i)T^{2} \)
41 \( 1 + (0.562 + 7.50i)T + (-40.5 + 6.11i)T^{2} \)
43 \( 1 + (-0.202 + 2.70i)T + (-42.5 - 6.40i)T^{2} \)
47 \( 1 + (11.7 + 1.77i)T + (44.9 + 13.8i)T^{2} \)
53 \( 1 + (-8.56 + 7.95i)T + (3.96 - 52.8i)T^{2} \)
59 \( 1 + (-0.445 + 5.93i)T + (-58.3 - 8.79i)T^{2} \)
61 \( 1 + (9.41 - 2.90i)T + (50.4 - 34.3i)T^{2} \)
67 \( 1 + (-0.984 + 1.70i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (0.687 + 3.01i)T + (-63.9 + 30.8i)T^{2} \)
73 \( 1 + (4.13 - 10.5i)T + (-53.5 - 49.6i)T^{2} \)
79 \( 1 + (6.94 + 12.0i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-6.09 + 0.917i)T + (79.3 - 24.4i)T^{2} \)
89 \( 1 + (-1.77 + 0.267i)T + (85.0 - 26.2i)T^{2} \)
97 \( 1 + (-9.24 + 16.0i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.11225802216692209920203648487, −8.900950359576378830825689578661, −8.648646735140632393188982227177, −7.31694015394717202262160011370, −6.38991930998280568237808859968, −5.27335056186881033657903926321, −4.81500907938102888855010344770, −3.66544116821445260274081526802, −3.09297796743288015416692141193, −0.49078914447189947212867064991, 1.36619533639391522395011529355, 3.02538833798460607649714571738, 3.92922042579159147997303698820, 4.74781394171713859662297500851, 6.14520788651403087142635445580, 6.87416275008875844609219737021, 7.58816383053474925497431286278, 7.892821095196135813327508123263, 9.800360093683792287421372819811, 10.40785129790905496164829488461

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