Properties

Label 2-882-7.6-c4-0-26
Degree $2$
Conductor $882$
Sign $0.755 - 0.654i$
Analytic cond. $91.1723$
Root an. cond. $9.54841$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.82·2-s + 8.00·4-s + 25.3i·5-s − 22.6·8-s − 71.8i·10-s − 43.9·11-s + 162. i·13-s + 64.0·16-s − 108. i·17-s − 449. i·19-s + 203. i·20-s + 124.·22-s + 435.·23-s − 19.4·25-s − 459. i·26-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.500·4-s + 1.01i·5-s − 0.353·8-s − 0.718i·10-s − 0.363·11-s + 0.961i·13-s + 0.250·16-s − 0.374i·17-s − 1.24i·19-s + 0.507i·20-s + 0.256·22-s + 0.823·23-s − 0.0311·25-s − 0.679i·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.755 - 0.654i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.755 - 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.755 - 0.654i$
Analytic conductor: \(91.1723\)
Root analytic conductor: \(9.54841\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{882} (685, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 882,\ (\ :2),\ 0.755 - 0.654i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.388595860\)
\(L(\frac12)\) \(\approx\) \(1.388595860\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 2.82T \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 - 25.3iT - 625T^{2} \)
11 \( 1 + 43.9T + 1.46e4T^{2} \)
13 \( 1 - 162. iT - 2.85e4T^{2} \)
17 \( 1 + 108. iT - 8.35e4T^{2} \)
19 \( 1 + 449. iT - 1.30e5T^{2} \)
23 \( 1 - 435.T + 2.79e5T^{2} \)
29 \( 1 + 742.T + 7.07e5T^{2} \)
31 \( 1 + 1.03e3iT - 9.23e5T^{2} \)
37 \( 1 - 986.T + 1.87e6T^{2} \)
41 \( 1 - 1.14e3iT - 2.82e6T^{2} \)
43 \( 1 - 2.41e3T + 3.41e6T^{2} \)
47 \( 1 + 1.34e3iT - 4.87e6T^{2} \)
53 \( 1 - 3.56e3T + 7.89e6T^{2} \)
59 \( 1 + 4.28e3iT - 1.21e7T^{2} \)
61 \( 1 - 1.39e3iT - 1.38e7T^{2} \)
67 \( 1 - 7.26e3T + 2.01e7T^{2} \)
71 \( 1 + 5.98e3T + 2.54e7T^{2} \)
73 \( 1 + 577. iT - 2.83e7T^{2} \)
79 \( 1 + 3.14e3T + 3.89e7T^{2} \)
83 \( 1 - 4.72e3iT - 4.74e7T^{2} \)
89 \( 1 + 837. iT - 6.27e7T^{2} \)
97 \( 1 - 5.62e3iT - 8.85e7T^{2} \)
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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

−9.531817582711363365771665631372, −9.038715116002458039082426162700, −7.88779398094189097768973086753, −7.07645446199653435660002037215, −6.59422935798486340018437888828, −5.43087866380397271117764288696, −4.19220407093578355025768451577, −2.91752497365027447784724972894, −2.20185951989549439389420992902, −0.68683634125390101102052429953, 0.62307426048464318924639486627, 1.50999558459574949783856015422, 2.83967398720702717457000692203, 4.04392028461317290313097631270, 5.27434640911826518865529906378, 5.87584636097246163066757767760, 7.17932888611649421642343174525, 7.964476374129798523053732106429, 8.642116371034383262395459396300, 9.338839647376283166231064374636

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