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Properties

Label 2-882-63.25-c1-0-39
Degree $2$
Conductor $882$
Sign $0.351 + 0.936i$
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 + (1.29 − 1.15i)3-s + 4-s + (1.84 − 3.20i)5-s + (1.29 − 1.15i)6-s + 8-s + (0.349 − 2.97i)9-s + (1.84 − 3.20i)10-s + (0.738 + 1.27i)11-s + (1.29 − 1.15i)12-s + (1.34 + 2.33i)13-s + (−1.29 − 6.27i)15-s + 16-s + (−3.28 + 5.69i)17-s + (0.349 − 2.97i)18-s + (0.444 + 0.769i)19-s + ⋯
L(s)  = 1  + 0.707·2-s + (0.747 − 0.664i)3-s + 0.5·4-s + (0.827 − 1.43i)5-s + (0.528 − 0.469i)6-s + 0.353·8-s + (0.116 − 0.993i)9-s + (0.584 − 1.01i)10-s + (0.222 + 0.385i)11-s + (0.373 − 0.332i)12-s + (0.374 + 0.648i)13-s + (−0.334 − 1.62i)15-s + 0.250·16-s + (−0.797 + 1.38i)17-s + (0.0824 − 0.702i)18-s + (0.101 + 0.176i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.351 + 0.936i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.351 + 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.351 + 0.936i$
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: \((2,\ 882,\ (\ :1/2),\ 0.351 + 0.936i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.86552 - 1.98544i\)
\(L(\frac12)\) \(\approx\) \(2.86552 - 1.98544i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.29 + 1.15i)T \)
7 \( 1 \)
good5 \( 1 + (-1.84 + 3.20i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-0.738 - 1.27i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.34 - 2.33i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (3.28 - 5.69i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.444 - 0.769i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.14 - 5.44i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.25 + 2.17i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 6.81T + 31T^{2} \)
37 \( 1 + (1.38 + 2.40i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-2.05 - 3.56i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.00618 + 0.0107i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 6.98T + 47T^{2} \)
53 \( 1 + (1.60 - 2.78i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 6.90T + 59T^{2} \)
61 \( 1 - 5.73T + 61T^{2} \)
67 \( 1 + 9.46T + 67T^{2} \)
71 \( 1 + 5.46T + 71T^{2} \)
73 \( 1 + (-6.03 + 10.4i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 - 11.4T + 79T^{2} \)
83 \( 1 + (2.23 - 3.87i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-4.43 - 7.68i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-6.58 + 11.4i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.667797247907755524035850265567, −9.096941107790105936141255645568, −8.354167851475460577457884870851, −7.42809558042977663756704006708, −6.29254077905665544455301675051, −5.72852875251323530813388814971, −4.47947976203693375101802488878, −3.72909827471730383056553610053, −2.05594291034276303357457543887, −1.49937566032210452523044359072, 2.23121504244505386077599444888, 2.90075641032200970439542504628, 3.75501616035043568760021663945, 4.95202955347976967526488487505, 5.91135478785735561294195617313, 6.79777861505703613728386540166, 7.56489612190001040454803851766, 8.766550146547560065018069844375, 9.563646042107216467601671270288, 10.57921411281153595621442231640

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