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Properties

Label 2-882-63.38-c1-0-30
Degree $2$
Conductor $882$
Sign $-0.925 + 0.378i$
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  − i·2-s + (−1.64 + 0.533i)3-s − 4-s + (0.450 − 0.779i)5-s + (0.533 + 1.64i)6-s + i·8-s + (2.43 − 1.75i)9-s + (−0.779 − 0.450i)10-s + (−2.70 + 1.56i)11-s + (1.64 − 0.533i)12-s + (1.99 − 1.14i)13-s + (−0.325 + 1.52i)15-s + 16-s + (2.57 − 4.46i)17-s + (−1.75 − 2.43i)18-s + (−2.38 + 1.37i)19-s + ⋯
L(s)  = 1  − 0.707i·2-s + (−0.951 + 0.308i)3-s − 0.5·4-s + (0.201 − 0.348i)5-s + (0.217 + 0.672i)6-s + 0.353i·8-s + (0.810 − 0.586i)9-s + (−0.246 − 0.142i)10-s + (−0.816 + 0.471i)11-s + (0.475 − 0.154i)12-s + (0.552 − 0.318i)13-s + (−0.0840 + 0.393i)15-s + 0.250·16-s + (0.624 − 1.08i)17-s + (−0.414 − 0.572i)18-s + (−0.546 + 0.315i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.925 + 0.378i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.925 + 0.378i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.925 + 0.378i$
Analytic conductor: \(7.04280\)
Root analytic conductor: \(2.65382\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{882} (227, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 882,\ (\ :1/2),\ -0.925 + 0.378i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.120452 - 0.611951i\)
\(L(\frac12)\) \(\approx\) \(0.120452 - 0.611951i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
3 \( 1 + (1.64 - 0.533i)T \)
7 \( 1 \)
good5 \( 1 + (-0.450 + 0.779i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (2.70 - 1.56i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.99 + 1.14i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2.57 + 4.46i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.38 - 1.37i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.48 + 0.857i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.85 + 1.07i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + 10.0iT - 31T^{2} \)
37 \( 1 + (4.73 + 8.20i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.22 + 2.11i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.273 - 0.473i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 7.86T + 47T^{2} \)
53 \( 1 + (12.0 + 6.97i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + 7.98T + 59T^{2} \)
61 \( 1 - 7.25iT - 61T^{2} \)
67 \( 1 - 3.67T + 67T^{2} \)
71 \( 1 + 14.1iT - 71T^{2} \)
73 \( 1 + (-10.9 - 6.30i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + 6.54T + 79T^{2} \)
83 \( 1 + (0.184 - 0.319i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (6.00 + 10.3i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (8.86 + 5.12i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.825561934658486595874359870855, −9.371661100206175043528253647441, −8.141275303623586394489950248022, −7.22489082531152936252173599229, −5.96101006011540603628089457694, −5.27206543779350523645093740396, −4.45069466964600462173675911624, −3.34256105792489354738279932065, −1.86040305608286779158313055870, −0.34986114822181402267281232332, 1.51162365197038824166902706227, 3.27169480226811902932144094873, 4.61211458104703534502725295046, 5.43523310336207872485168233831, 6.33610241417324852705400738434, 6.76434944223371001566727062296, 7.960720575751136185832011259064, 8.497819109768917588975438511729, 9.822262601394817769593226625596, 10.58748811194000684304452830614

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