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Properties

Label 2-882-63.16-c1-0-33
Degree $2$
Conductor $882$
Sign $0.0788 + 0.996i$
Analytic cond. $7.04280$
Root an. cond. $2.65382$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 + 0.866i)2-s + 1.73i·3-s + (−0.499 + 0.866i)4-s − 2·5-s + (−1.49 + 0.866i)6-s − 0.999·8-s − 2.99·9-s + (−1 − 1.73i)10-s + 11-s + (−1.49 − 0.866i)12-s + (−3 − 5.19i)13-s − 3.46i·15-s + (−0.5 − 0.866i)16-s + (−2.5 − 4.33i)17-s + (−1.49 − 2.59i)18-s + (−3.5 + 6.06i)19-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + 0.999i·3-s + (−0.249 + 0.433i)4-s − 0.894·5-s + (−0.612 + 0.353i)6-s − 0.353·8-s − 0.999·9-s + (−0.316 − 0.547i)10-s + 0.301·11-s + (−0.433 − 0.249i)12-s + (−0.832 − 1.44i)13-s − 0.894i·15-s + (−0.125 − 0.216i)16-s + (−0.606 − 1.05i)17-s + (−0.353 − 0.612i)18-s + (−0.802 + 1.39i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0788 + 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0788 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.0788 + 0.996i$
Analytic conductor: \(7.04280\)
Root analytic conductor: \(2.65382\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{882} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 882,\ (\ :1/2),\ 0.0788 + 0.996i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.73iT \)
7 \( 1 \)
good5 \( 1 + 2T + 5T^{2} \)
11 \( 1 - T + 11T^{2} \)
13 \( 1 + (3 + 5.19i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (2.5 + 4.33i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.5 - 6.06i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 - 4T + 23T^{2} \)
29 \( 1 + (-2 + 3.46i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (3 - 5.19i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (1 - 1.73i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-1.5 - 2.59i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.5 + 0.866i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (6 + 10.3i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (3.5 - 6.06i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (6 + 10.3i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.5 - 11.2i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 8T + 71T^{2} \)
73 \( 1 + (-0.5 - 0.866i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-3 - 5.19i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-8 + 13.8i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (3 - 5.19i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (2.5 - 4.33i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.940396021989886035491900777975, −8.997725958452179417790194004687, −8.153032747296355867613584829145, −7.53330820577589880628333198545, −6.38357716082754974938204037881, −5.33373518892321937314664540923, −4.63919110831609025510858491016, −3.71174488859932415149020344677, −2.84575367638246797826555625003, 0, 1.70077269937731846127516282317, 2.71572073857707648349668193531, 4.03053458635514802427630038701, 4.77276804566385830257117393465, 6.17537932399537230252458479174, 6.90918376905108644980201328654, 7.66789065723497343745787176113, 8.834762969766625512755992773921, 9.232718206710121593931840859865

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