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Properties

Label 2-882-63.25-c1-0-34
Degree $2$
Conductor $882$
Sign $-0.888 + 0.458i$
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 − 1.73i·3-s + 4-s + (1.5 − 2.59i)5-s + 1.73i·6-s − 8-s − 2.99·9-s + (−1.5 + 2.59i)10-s + (1.5 + 2.59i)11-s − 1.73i·12-s + (−0.5 − 0.866i)13-s + (−4.5 − 2.59i)15-s + 16-s + (1.5 − 2.59i)17-s + 2.99·18-s + (−3.5 − 6.06i)19-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.999i·3-s + 0.5·4-s + (0.670 − 1.16i)5-s + 0.707i·6-s − 0.353·8-s − 0.999·9-s + (−0.474 + 0.821i)10-s + (0.452 + 0.783i)11-s − 0.499i·12-s + (−0.138 − 0.240i)13-s + (−1.16 − 0.670i)15-s + 0.250·16-s + (0.363 − 0.630i)17-s + 0.707·18-s + (−0.802 − 1.39i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.888 + 0.458i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.888 + 0.458i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.888 + 0.458i$
Analytic conductor: \(7.04280\)
Root analytic conductor: \(2.65382\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{882} (655, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 882,\ (\ :1/2),\ -0.888 + 0.458i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.245382 - 1.01188i\)
\(L(\frac12)\) \(\approx\) \(0.245382 - 1.01188i\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.73iT \)
7 \( 1 \)
good5 \( 1 + (-1.5 + 2.59i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1.5 - 2.59i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.5 + 0.866i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-1.5 + 2.59i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.5 + 6.06i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-4.5 + 7.79i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.5 - 2.59i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 8T + 31T^{2} \)
37 \( 1 + (-0.5 - 0.866i)T + (-18.5 + 32.0i)T^{2} \)
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 \( 1 + 47T^{2} \)
53 \( 1 + (1.5 - 2.59i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 + 2T + 61T^{2} \)
67 \( 1 + 4T + 67T^{2} \)
71 \( 1 - 12T + 71T^{2} \)
73 \( 1 + (-5.5 + 9.52i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + 16T + 79T^{2} \)
83 \( 1 + (4.5 - 7.79i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-1.5 - 2.59i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (0.5 - 0.866i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.330575465001104408978983883949, −9.063372790816131529363498390117, −8.230039808944002442357392488973, −7.17869515237098545801502589486, −6.62053404230005211540171888006, −5.48515262991420957884108594787, −4.65000065925864479939546635242, −2.74438599493495838160807960792, −1.74079645369750714462172359529, −0.62175685291960500621511822822, 1.86662004822149684102865916908, 3.17609009421795780705131170886, 3.87426043143048719017810797897, 5.60617276254181397099908486755, 6.04688242461541621953646227894, 7.11108174311981492139832303749, 8.137300721426646937940238490548, 9.057205605416533279803614600937, 9.729209829164765977743811430845, 10.39562127406394720345447818060

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