Properties

Label 2-882-7.2-c3-0-34
Degree $2$
Conductor $882$
Sign $-0.0725 + 0.997i$
Analytic cond. $52.0396$
Root an. cond. $7.21385$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1 − 1.73i)2-s + (−1.99 − 3.46i)4-s + (−3.70 + 6.42i)5-s − 7.99·8-s + (7.41 + 12.8i)10-s + (5.24 + 9.08i)11-s − 2.78·13-s + (−8 + 13.8i)16-s + (−25.2 − 43.6i)17-s + (62.5 − 108. i)19-s + 29.6·20-s + 20.9·22-s + (−91.1 + 157. i)23-s + (35.0 + 60.6i)25-s + (−2.78 + 4.82i)26-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.331 + 0.574i)5-s − 0.353·8-s + (0.234 + 0.406i)10-s + (0.143 + 0.248i)11-s − 0.0594·13-s + (−0.125 + 0.216i)16-s + (−0.359 − 0.623i)17-s + (0.754 − 1.30i)19-s + 0.331·20-s + 0.203·22-s + (−0.826 + 1.43i)23-s + (0.280 + 0.485i)25-s + (−0.0210 + 0.0364i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0725 + 0.997i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.0725 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.0725 + 0.997i$
Analytic conductor: \(52.0396\)
Root analytic conductor: \(7.21385\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{882} (667, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 882,\ (\ :3/2),\ -0.0725 + 0.997i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.839456992\)
\(L(\frac12)\) \(\approx\) \(1.839456992\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1 + 1.73i)T \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (3.70 - 6.42i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (-5.24 - 9.08i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + 2.78T + 2.19e3T^{2} \)
17 \( 1 + (25.2 + 43.6i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-62.5 + 108. i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (91.1 - 157. i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + 156.T + 2.43e4T^{2} \)
31 \( 1 + (-69.8 - 120. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (-197. + 341. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 - 197.T + 6.89e4T^{2} \)
43 \( 1 - 343.T + 7.95e4T^{2} \)
47 \( 1 + (-305. + 528. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (68.7 + 119. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (294. + 510. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-123. + 214. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-197. - 342. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 285.T + 3.57e5T^{2} \)
73 \( 1 + (498. + 863. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-424. + 734. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + 210.T + 5.71e5T^{2} \)
89 \( 1 + (-276. + 479. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 - 903.T + 9.12e5T^{2} \)
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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

−9.476274533731288961265701352180, −9.077878082509558782824508303494, −7.57571911582057493960514755030, −7.12026434789838243018859098587, −5.88668546350270029451331823472, −4.99629848373123895943122942232, −3.93498955366273542526212118252, −3.05920866239649841382141776416, −2.00703522574291181422174451336, −0.51197566317782901296587770156, 0.991519659398431442247503368102, 2.60778978049516461628672233542, 3.99438367542033074603100765357, 4.50670973592099832736523013933, 5.80484605841486410191190142055, 6.28182422634292639031004839689, 7.59184103924856790516689173433, 8.122635586624314908598874949837, 8.927329131085726933048040968818, 9.850165804431666084762628517775

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