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Properties

Label 2-882-63.58-c1-0-38
Degree $2$
Conductor $882$
Sign $0.101 + 0.994i$
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 + (1.67 − 0.448i)3-s + 4-s + (−1.93 − 3.34i)5-s + (1.67 − 0.448i)6-s + 8-s + (2.59 − 1.50i)9-s + (−1.93 − 3.34i)10-s + (1.86 − 3.23i)11-s + (1.67 − 0.448i)12-s + (−3.34 + 5.79i)13-s + (−4.73 − 4.73i)15-s + 16-s + (−2.70 − 4.69i)17-s + (2.59 − 1.50i)18-s + (1.48 − 2.56i)19-s + ⋯
L(s)  = 1  + 0.707·2-s + (0.965 − 0.258i)3-s + 0.5·4-s + (−0.863 − 1.49i)5-s + (0.683 − 0.183i)6-s + 0.353·8-s + (0.866 − 0.5i)9-s + (−0.610 − 1.05i)10-s + (0.562 − 0.974i)11-s + (0.482 − 0.129i)12-s + (−0.928 + 1.60i)13-s + (−1.22 − 1.22i)15-s + 0.250·16-s + (−0.656 − 1.13i)17-s + (0.612 − 0.353i)18-s + (0.340 − 0.589i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.101 + 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.101 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.101 + 0.994i$
Analytic conductor: \(7.04280\)
Root analytic conductor: \(2.65382\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{882} (373, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 882,\ (\ :1/2),\ 0.101 + 0.994i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.10639 - 1.90198i\)
\(L(\frac12)\) \(\approx\) \(2.10639 - 1.90198i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.67 + 0.448i)T \)
7 \( 1 \)
good5 \( 1 + (1.93 + 3.34i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.86 + 3.23i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (3.34 - 5.79i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (2.70 + 4.69i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.48 + 2.56i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.732 - 1.26i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2 - 3.46i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 1.79T + 31T^{2} \)
37 \( 1 + (0.267 - 0.464i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-0.637 + 1.10i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.86 + 3.23i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 10.5T + 47T^{2} \)
53 \( 1 + (-1.46 - 2.53i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + 8.62T + 59T^{2} \)
61 \( 1 - 6.96T + 61T^{2} \)
67 \( 1 - 5.53T + 67T^{2} \)
71 \( 1 - 2.53T + 71T^{2} \)
73 \( 1 + (-3.41 - 5.91i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + 4.92T + 79T^{2} \)
83 \( 1 + (-8.95 - 15.5i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (3.53 - 6.12i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-2.94 - 5.10i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.453017988187968232246766461603, −9.079996349321342669531664922538, −8.353458700947837892957355458171, −7.31940244262237887738708543207, −6.72706999113755206335641963601, −5.16987537964599831707052959682, −4.44841249142198992604497987155, −3.72806045049423439004295266105, −2.44759619915682041507442755616, −1.02547229819349956648295903455, 2.21394590791337583952245270181, 3.05491172677448435731186771378, 3.84838826292867899419254302013, 4.66365594389083218339407073655, 6.11803549730302934231448101978, 7.10004142973133102918020343324, 7.63287185894135904784011327284, 8.355317423674353064828489039009, 9.824286645203787523751053696013, 10.33499664557556920511623270357

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