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Properties

Label 2-882-441.16-c1-0-46
Degree $2$
Conductor $882$
Sign $0.997 - 0.0717i$
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 + (1.52 − 0.829i)3-s + (0.955 + 0.294i)4-s + (1.93 + 0.929i)5-s + (1.62 − 0.593i)6-s + (1.37 + 2.25i)7-s + (0.900 + 0.433i)8-s + (1.62 − 2.52i)9-s + (1.77 + 1.20i)10-s + (1.91 − 2.40i)11-s + (1.69 − 0.344i)12-s + (−3.96 − 0.598i)13-s + (1.02 + 2.43i)14-s + (3.70 − 0.188i)15-s + (0.826 + 0.563i)16-s + (−6.85 + 2.11i)17-s + ⋯
L(s)  = 1  + (0.699 + 0.105i)2-s + (0.877 − 0.479i)3-s + (0.477 + 0.147i)4-s + (0.863 + 0.415i)5-s + (0.664 − 0.242i)6-s + (0.520 + 0.853i)7-s + (0.318 + 0.153i)8-s + (0.540 − 0.841i)9-s + (0.559 + 0.381i)10-s + (0.578 − 0.725i)11-s + (0.489 − 0.0995i)12-s + (−1.10 − 0.165i)13-s + (0.274 + 0.651i)14-s + (0.957 − 0.0486i)15-s + (0.206 + 0.140i)16-s + (−1.66 + 0.513i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0717i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 - 0.0717i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.997 - 0.0717i$
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: \((2,\ 882,\ (\ :1/2),\ 0.997 - 0.0717i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.69455 + 0.132773i\)
\(L(\frac12)\) \(\approx\) \(3.69455 + 0.132773i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.52 + 0.829i)T \)
7 \( 1 + (-1.37 - 2.25i)T \)
good5 \( 1 + (-1.93 - 0.929i)T + (3.11 + 3.90i)T^{2} \)
11 \( 1 + (-1.91 + 2.40i)T + (-2.44 - 10.7i)T^{2} \)
13 \( 1 + (3.96 + 0.598i)T + (12.4 + 3.83i)T^{2} \)
17 \( 1 + (6.85 - 2.11i)T + (14.0 - 9.57i)T^{2} \)
19 \( 1 + (-0.586 + 1.01i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.49 - 6.57i)T + (-20.7 + 9.97i)T^{2} \)
29 \( 1 + (6.37 + 5.91i)T + (2.16 + 28.9i)T^{2} \)
31 \( 1 + (-0.810 + 1.40i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.692 + 0.642i)T + (2.76 + 36.8i)T^{2} \)
41 \( 1 + (-0.595 - 7.94i)T + (-40.5 + 6.11i)T^{2} \)
43 \( 1 + (0.381 - 5.08i)T + (-42.5 - 6.40i)T^{2} \)
47 \( 1 + (4.27 + 0.643i)T + (44.9 + 13.8i)T^{2} \)
53 \( 1 + (-9.09 + 8.44i)T + (3.96 - 52.8i)T^{2} \)
59 \( 1 + (-0.962 + 12.8i)T + (-58.3 - 8.79i)T^{2} \)
61 \( 1 + (-7.88 + 2.43i)T + (50.4 - 34.3i)T^{2} \)
67 \( 1 + (-0.579 + 1.00i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (2.20 + 9.66i)T + (-63.9 + 30.8i)T^{2} \)
73 \( 1 + (5.04 - 12.8i)T + (-53.5 - 49.6i)T^{2} \)
79 \( 1 + (3.99 + 6.92i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (6.63 - 1.00i)T + (79.3 - 24.4i)T^{2} \)
89 \( 1 + (-17.0 + 2.56i)T + (85.0 - 26.2i)T^{2} \)
97 \( 1 + (1.10 - 1.90i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.915219392004772217673343608044, −9.291832021059277205897576903609, −8.417281502034081418466861823990, −7.54634176965116583378292029100, −6.54524925717507825840529827073, −5.94596766265067135465460910418, −4.84754675604947322124303680079, −3.62221903520754787452746681793, −2.47751029983856646189768699034, −1.89868319145244104234009556363, 1.73364779800643158060635788942, 2.52141304057465964545089418975, 4.02853537067507749602390418006, 4.60367182425888358584139745835, 5.37462764886222662998064409688, 6.96579763062146860308801459306, 7.26171364350979343970097155627, 8.711234499319333766410073996637, 9.266344572566570295529214650186, 10.19201229143082736574733192362

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