Properties

Label 2-882-3.2-c2-0-7
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 + 1.41i·5-s − 2.82i·8-s − 2.00·10-s + 7.07i·11-s + 15·13-s + 4.00·16-s + 11.3i·17-s − 13·19-s − 2.82i·20-s − 10.0·22-s − 22.6i·23-s + 23·25-s + 21.2i·26-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.500·4-s + 0.282i·5-s − 0.353i·8-s − 0.200·10-s + 0.642i·11-s + 1.15·13-s + 0.250·16-s + 0.665i·17-s − 0.684·19-s − 0.141i·20-s − 0.454·22-s − 0.983i·23-s + 0.920·25-s + 0.815i·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

\(L(\frac{3}{2})\) \(\approx\) \(1.426884238\)
\(L(\frac12)\) \(\approx\) \(1.426884238\)
\(L(2)\) 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 \( 1 - 1.41iT - 25T^{2} \)
11 \( 1 - 7.07iT - 121T^{2} \)
13 \( 1 - 15T + 169T^{2} \)
17 \( 1 - 11.3iT - 289T^{2} \)
19 \( 1 + 13T + 361T^{2} \)
23 \( 1 + 22.6iT - 529T^{2} \)
29 \( 1 - 22.6iT - 841T^{2} \)
31 \( 1 - 3T + 961T^{2} \)
37 \( 1 - 17T + 1.36e3T^{2} \)
41 \( 1 - 80.6iT - 1.68e3T^{2} \)
43 \( 1 + 85T + 1.84e3T^{2} \)
47 \( 1 - 72.1iT - 2.20e3T^{2} \)
53 \( 1 + 33.9iT - 2.80e3T^{2} \)
59 \( 1 - 90.5iT - 3.48e3T^{2} \)
61 \( 1 + 72T + 3.72e3T^{2} \)
67 \( 1 - 43T + 4.48e3T^{2} \)
71 \( 1 + 52.3iT - 5.04e3T^{2} \)
73 \( 1 + 95T + 5.32e3T^{2} \)
79 \( 1 - 69T + 6.24e3T^{2} \)
83 \( 1 + 60.8iT - 6.88e3T^{2} \)
89 \( 1 - 135. iT - 7.92e3T^{2} \)
97 \( 1 - 16T + 9.40e3T^{2} \)
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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

−10.33483416973739514455699379715, −9.246404499572143332290508045652, −8.483446681638148964994144075885, −7.79454041246524013305767998474, −6.56689313010192626865148743191, −6.32571181011750943852912596866, −4.99037568207296011803907691133, −4.16355648055610461305929991480, −2.97975731402677073802145215034, −1.38706044404102170647020819007, 0.50210037649063395664577859525, 1.76748740565168009105784053164, 3.12239826401546163222934072125, 3.96546391951100975620875913794, 5.08831221549109123911256204597, 5.96909271287050820899040322624, 7.03183669353088332145770969657, 8.274869924013583988531479582796, 8.759400003704788231675182017771, 9.647689109089151303082928406159

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