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Properties

Label 2-882-21.17-c1-0-7
Degree $2$
Conductor $882$
Sign $0.360 + 0.932i$
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  + (0.866 − 0.5i)2-s + (0.499 − 0.866i)4-s + (0.358 + 0.621i)5-s − 0.999i·8-s + (0.621 + 0.358i)10-s + (−2.59 − 1.5i)11-s − 2.44i·13-s + (−0.5 − 0.866i)16-s + (2.95 − 5.12i)17-s + (5.12 − 2.95i)19-s + 0.717·20-s − 3·22-s + (3.67 − 2.12i)23-s + (2.24 − 3.88i)25-s + (−1.22 − 2.12i)26-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.249 − 0.433i)4-s + (0.160 + 0.277i)5-s − 0.353i·8-s + (0.196 + 0.113i)10-s + (−0.783 − 0.452i)11-s − 0.679i·13-s + (−0.125 − 0.216i)16-s + (0.717 − 1.24i)17-s + (1.17 − 0.678i)19-s + 0.160·20-s − 0.639·22-s + (0.766 − 0.442i)23-s + (0.448 − 0.776i)25-s + (−0.240 − 0.416i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.360 + 0.932i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.360 + 0.932i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.360 + 0.932i$
Analytic conductor: \(7.04280\)
Root analytic conductor: \(2.65382\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{882} (521, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 882,\ (\ :1/2),\ 0.360 + 0.932i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.83859 - 1.26116i\)
\(L(\frac12)\) \(\approx\) \(1.83859 - 1.26116i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 + 0.5i)T \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (-0.358 - 0.621i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (2.59 + 1.5i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 2.44iT - 13T^{2} \)
17 \( 1 + (-2.95 + 5.12i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-5.12 + 2.95i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.67 + 2.12i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 7.24iT - 29T^{2} \)
31 \( 1 + (-7.86 - 4.54i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.121 - 0.210i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 11.8T + 41T^{2} \)
43 \( 1 + 0.242T + 43T^{2} \)
47 \( 1 + (-2.95 - 5.12i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (6.27 + 3.62i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (4.03 - 6.98i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.878 - 0.507i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5 + 8.66i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 1.75iT - 71T^{2} \)
73 \( 1 + (-1.24 - 0.717i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-1.37 - 2.38i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 6.63T + 83T^{2} \)
89 \( 1 + (5.19 + 9i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 13.5iT - 97T^{2} \)
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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

−10.24062433754779692076627058106, −9.295574801885053761806374091264, −8.263884451627303118565819054531, −7.27691195901185472071867109155, −6.48320405342659234325864112330, −5.22228158821719320097595761893, −4.92916645895491898174357317851, −3.12407429221355645177440607439, −2.86980643062670198621531290564, −0.960530131540207451571737132879, 1.62194154475146910129968374014, 3.02568056876845092854215626503, 4.08386710965034839805966612249, 5.10823209775616378559687765090, 5.79883665586143555013064893796, 6.81087382196330350145051644640, 7.74307227790556910820021386612, 8.365963099109448740902075602661, 9.577567118168376174829807976908, 10.17336672628013998856832207092

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