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Properties

Label 2-882-21.11-c2-0-12
Degree $2$
Conductor $882$
Sign $0.645 + 0.763i$
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  + (1.22 − 0.707i)2-s + (0.999 − 1.73i)4-s + (−5.25 + 3.03i)5-s − 2.82i·8-s + (−4.29 + 7.43i)10-s + (−10.5 − 6.06i)11-s + 18.5·13-s + (−2.00 − 3.46i)16-s + (9.44 + 5.45i)17-s + (10 + 17.3i)19-s + 12.1i·20-s − 17.1·22-s + (10.5 − 6.06i)23-s + (5.91 − 10.2i)25-s + (22.7 − 13.1i)26-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.249 − 0.433i)4-s + (−1.05 + 0.606i)5-s − 0.353i·8-s + (−0.429 + 0.743i)10-s + (−0.955 − 0.551i)11-s + 1.42·13-s + (−0.125 − 0.216i)16-s + (0.555 + 0.320i)17-s + (0.526 + 0.911i)19-s + 0.606i·20-s − 0.780·22-s + (0.457 − 0.263i)23-s + (0.236 − 0.409i)25-s + (0.875 − 0.505i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.645 + 0.763i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.645 + 0.763i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.645 + 0.763i$
Analytic conductor: \(24.0327\)
Root analytic conductor: \(4.90232\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{882} (557, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 882,\ (\ :1),\ 0.645 + 0.763i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.168821619\)
\(L(\frac12)\) \(\approx\) \(2.168821619\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.22 + 0.707i)T \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (5.25 - 3.03i)T + (12.5 - 21.6i)T^{2} \)
11 \( 1 + (10.5 + 6.06i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 - 18.5T + 169T^{2} \)
17 \( 1 + (-9.44 - 5.45i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (-10 - 17.3i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (-10.5 + 6.06i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + 41.8iT - 841T^{2} \)
31 \( 1 + (-12.5 + 21.7i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + (19 + 32.9i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 + 60.6iT - 1.68e3T^{2} \)
43 \( 1 - 83.4T + 1.84e3T^{2} \)
47 \( 1 + (14.6 - 8.48i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-81.4 - 47.0i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (-50.4 - 29.1i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-7.83 - 13.5i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-66.3 + 114. i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 12.1iT - 5.04e3T^{2} \)
73 \( 1 + (38.4 - 66.6i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (16.8 + 29.1i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + 60.5iT - 6.88e3T^{2} \)
89 \( 1 + (4.13 - 2.38i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 - 188.T + 9.40e3T^{2} \)
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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

−10.23159145918633125358803906353, −8.912345130989729869388566471382, −7.954011124858503287696977849228, −7.39179812470736534191092338142, −6.10000898509037993223364400255, −5.53720418415285968330680789529, −4.03079395676893534197661381522, −3.59232947612648613872181638122, −2.47527730238463965127779859053, −0.74892871570755505121306216926, 1.03871051133808136101464744347, 2.89076019832318934431871084183, 3.79628154203259334053849593214, 4.82297934413963870810222813112, 5.41296266018348711101652435531, 6.71586868304173379182986716468, 7.47222400251826785696709698637, 8.284877108691160163068556983396, 8.898419616939291921048209378588, 10.14620908525760896169178297165

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