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Properties

Label 2-882-9.7-c1-0-25
Degree $2$
Conductor $882$
Sign $0.984 + 0.173i$
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  + (−0.5 + 0.866i)2-s + (1.67 − 0.448i)3-s + (−0.499 − 0.866i)4-s + (−0.517 − 0.896i)5-s + (−0.448 + 1.67i)6-s + 0.999·8-s + (2.59 − 1.50i)9-s + 1.03·10-s + (0.133 − 0.232i)11-s + (−1.22 − 1.22i)12-s + (0.896 + 1.55i)13-s + (−1.26 − 1.26i)15-s + (−0.5 + 0.866i)16-s + 6.83·17-s + 3i·18-s − 4.38·19-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (0.965 − 0.258i)3-s + (−0.249 − 0.433i)4-s + (−0.231 − 0.400i)5-s + (−0.183 + 0.683i)6-s + 0.353·8-s + (0.866 − 0.5i)9-s + 0.327·10-s + (0.0403 − 0.0699i)11-s + (−0.353 − 0.353i)12-s + (0.248 + 0.430i)13-s + (−0.327 − 0.327i)15-s + (−0.125 + 0.216i)16-s + 1.65·17-s + 0.707i·18-s − 1.00·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 + 0.173i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.984 + 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.984 + 0.173i$
Analytic conductor: \(7.04280\)
Root analytic conductor: \(2.65382\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{882} (295, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 882,\ (\ :1/2),\ 0.984 + 0.173i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.78693 - 0.156336i\)
\(L(\frac12)\) \(\approx\) \(1.78693 - 0.156336i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 - 0.866i)T \)
3 \( 1 + (-1.67 + 0.448i)T \)
7 \( 1 \)
good5 \( 1 + (0.517 + 0.896i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-0.133 + 0.232i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.896 - 1.55i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 - 6.83T + 17T^{2} \)
19 \( 1 + 4.38T + 19T^{2} \)
23 \( 1 + (2.73 + 4.73i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2 + 3.46i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-3.34 - 5.79i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 7.46T + 37T^{2} \)
41 \( 1 + (4.31 + 7.46i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.133 - 0.232i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-0.378 + 0.656i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 10.9T + 53T^{2} \)
59 \( 1 + (0.637 + 1.10i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (6.31 - 10.9i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.23 + 10.7i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 9.46T + 71T^{2} \)
73 \( 1 + 5.41T + 73T^{2} \)
79 \( 1 + (4.46 - 7.73i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-3.29 + 5.70i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 7.07T + 89T^{2} \)
97 \( 1 + (9.07 - 15.7i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.988711328871733158578899213816, −8.949302618833857469756747995883, −8.403707127795929533467067540980, −7.79772019654356899591627687217, −6.81572999901625512168922493733, −6.02242126627355054561624190409, −4.68721542192707001281038600980, −3.83187980918785329076372693431, −2.46685769890705190774969454148, −1.02627070929023879854352178936, 1.42532455514071576235778048113, 2.77929783830208376671381606112, 3.49157253478513465921269975493, 4.43378260592280982442702621276, 5.74127957784839920706304109189, 7.10073843310394166778380607117, 7.909047401201413277450907830821, 8.431417717481008552259409460131, 9.537996684850981875770359942997, 10.00908693929216787737949299373

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