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Properties

Label 2-882-63.25-c1-0-14
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 + (2.5 + 4.33i)13-s + (4.5 + 2.59i)15-s + 16-s + (1.5 − 2.59i)17-s − 2.99·18-s + (2.5 + 4.33i)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 + (0.693 + 1.20i)13-s + (1.16 + 0.670i)15-s + 0.250·16-s + (0.363 − 0.630i)17-s − 0.707·18-s + (0.573 + 0.993i)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

\(L(1)\) \(\approx\) \(2.08740 + 0.506197i\)
\(L(\frac12)\) \(\approx\) \(2.08740 + 0.506197i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.5 - 4.33i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-1.5 + 2.59i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.5 - 4.33i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.5 + 2.59i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.5 + 2.59i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 4T + 31T^{2} \)
37 \( 1 + (-3.5 - 6.06i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (4.5 + 7.79i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (5.5 - 9.52i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + (-1.5 + 2.59i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 12T + 59T^{2} \)
61 \( 1 + 2T + 61T^{2} \)
67 \( 1 + 4T + 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + (-5.5 + 9.52i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 - 8T + 79T^{2} \)
83 \( 1 + (-1.5 + 2.59i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-7.5 - 12.9i)T + (-44.5 + 77.0i)T^{2} \)
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

−10.45894240057243215525564160585, −9.378136630421024885442993889799, −8.126784336166302348008962832565, −7.44056363180921246131360944311, −6.63151702010508978452651763437, −6.27251888914789782586974806250, −4.81978626929117199288236542778, −3.68633543393850724388050810456, −2.80975501401100818542403723629, −1.57491184147247087469499887191, 0.907574533911859653391352160720, 3.10818549786906463443512333836, 3.74504527436505319337060098467, 4.75140182761791417006654958169, 5.39726015866936622502436385649, 6.24461950523342558662833986159, 7.73781825282072337968316940521, 8.500307128074790691881502643838, 9.088659824651638186147102741117, 10.21247363763923301307201137892

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