Properties

Label 2-882-3.2-c2-0-8
Degree $2$
Conductor $882$
Sign $-0.816 - 0.577i$
Analytic cond. $24.0327$
Root an. cond. $4.90232$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.41i·2-s − 2.00·4-s + 4i·5-s − 2.82i·8-s − 5.65·10-s − 2.82i·11-s + 12.7·13-s + 4.00·16-s + 4i·17-s + 22.6·19-s − 8.00i·20-s + 4.00·22-s + 36.7i·23-s + 9·25-s + 18i·26-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.500·4-s + 0.800i·5-s − 0.353i·8-s − 0.565·10-s − 0.257i·11-s + 0.979·13-s + 0.250·16-s + 0.235i·17-s + 1.19·19-s − 0.400i·20-s + 0.181·22-s + 1.59i·23-s + 0.359·25-s + 0.692i·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.816 - 0.577i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.816 - 0.577i$
Analytic conductor: \(24.0327\)
Root analytic conductor: \(4.90232\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{882} (197, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 882,\ (\ :1),\ -0.816 - 0.577i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.561681199\)
\(L(\frac12)\) \(\approx\) \(1.561681199\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 1.41iT \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 - 4iT - 25T^{2} \)
11 \( 1 + 2.82iT - 121T^{2} \)
13 \( 1 - 12.7T + 169T^{2} \)
17 \( 1 - 4iT - 289T^{2} \)
19 \( 1 - 22.6T + 361T^{2} \)
23 \( 1 - 36.7iT - 529T^{2} \)
29 \( 1 - 32.5iT - 841T^{2} \)
31 \( 1 + 50.9T + 961T^{2} \)
37 \( 1 + 32T + 1.36e3T^{2} \)
41 \( 1 + 38iT - 1.68e3T^{2} \)
43 \( 1 - 20T + 1.84e3T^{2} \)
47 \( 1 + 20iT - 2.20e3T^{2} \)
53 \( 1 - 94.7iT - 2.80e3T^{2} \)
59 \( 1 - 4iT - 3.48e3T^{2} \)
61 \( 1 - 83.4T + 3.72e3T^{2} \)
67 \( 1 + 48T + 4.48e3T^{2} \)
71 \( 1 - 76.3iT - 5.04e3T^{2} \)
73 \( 1 + 120.T + 5.32e3T^{2} \)
79 \( 1 + 148T + 6.24e3T^{2} \)
83 \( 1 - 80iT - 6.88e3T^{2} \)
89 \( 1 + 106iT - 7.92e3T^{2} \)
97 \( 1 - 154.T + 9.40e3T^{2} \)
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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

−10.30223366431066712669652741204, −9.221583154837629611110269202067, −8.638354291851642065026673946212, −7.38831352606443878770958447374, −7.10278913808860247598617952721, −5.87345101926073778281718802119, −5.35415218728470933272832447011, −3.84254587389051184293736457857, −3.15020572701284162199747998256, −1.38194832148221380729769100428, 0.56193555964805536961257468428, 1.71212627479738322386044140685, 3.05492151474207532749374526011, 4.13768526888750765063140877640, 4.99627272396817247424165704985, 5.91528427975366815224190816276, 7.10662046373853454793794181968, 8.215997516169116827841133213764, 8.848653056588670877399564576852, 9.610307458970324435192446740254

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