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Properties

Label 2-882-63.58-c1-0-18
Degree $2$
Conductor $882$
Sign $0.623 - 0.781i$
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  + 2-s + (−1.67 + 0.448i)3-s + 4-s + (1.93 + 3.34i)5-s + (−1.67 + 0.448i)6-s + 8-s + (2.59 − 1.50i)9-s + (1.93 + 3.34i)10-s + (1.86 − 3.23i)11-s + (−1.67 + 0.448i)12-s + (3.34 − 5.79i)13-s + (−4.73 − 4.73i)15-s + 16-s + (2.70 + 4.69i)17-s + (2.59 − 1.50i)18-s + (−1.48 + 2.56i)19-s + ⋯
L(s)  = 1  + 0.707·2-s + (−0.965 + 0.258i)3-s + 0.5·4-s + (0.863 + 1.49i)5-s + (−0.683 + 0.183i)6-s + 0.353·8-s + (0.866 − 0.5i)9-s + (0.610 + 1.05i)10-s + (0.562 − 0.974i)11-s + (−0.482 + 0.129i)12-s + (0.928 − 1.60i)13-s + (−1.22 − 1.22i)15-s + 0.250·16-s + (0.656 + 1.13i)17-s + (0.612 − 0.353i)18-s + (−0.340 + 0.589i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.623 - 0.781i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.623 - 0.781i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.623 - 0.781i$
Analytic conductor: \(7.04280\)
Root analytic conductor: \(2.65382\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{882} (373, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 882,\ (\ :1/2),\ 0.623 - 0.781i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.03296 + 0.978913i\)
\(L(\frac12)\) \(\approx\) \(2.03296 + 0.978913i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 + (1.67 - 0.448i)T \)
7 \( 1 \)
good5 \( 1 + (-1.93 - 3.34i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.86 + 3.23i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-3.34 + 5.79i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2.70 - 4.69i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.48 - 2.56i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.732 - 1.26i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2 - 3.46i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 1.79T + 31T^{2} \)
37 \( 1 + (0.267 - 0.464i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.637 - 1.10i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.86 + 3.23i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 10.5T + 47T^{2} \)
53 \( 1 + (-1.46 - 2.53i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 8.62T + 59T^{2} \)
61 \( 1 + 6.96T + 61T^{2} \)
67 \( 1 - 5.53T + 67T^{2} \)
71 \( 1 - 2.53T + 71T^{2} \)
73 \( 1 + (3.41 + 5.91i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + 4.92T + 79T^{2} \)
83 \( 1 + (8.95 + 15.5i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-3.53 + 6.12i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (2.94 + 5.10i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.45954559545608798999132481028, −9.989709728101438548633114677811, −8.505755626992545171005942245685, −7.36824059475691135330299889999, −6.20286857286500788167509872149, −6.12625795946792418949553750196, −5.30570888931168715737525271309, −3.65361718003387974963259365383, −3.21637288951050862899342757740, −1.48543453054798297276431184004, 1.18370013757149076263716823460, 2.05326011883508828501889527970, 4.18699049159293504617610676629, 4.74607521641751937160569201862, 5.48687926608263649027730822396, 6.45477565536761209812729318389, 7.01080113945374452067046098898, 8.380276034979767713615271730239, 9.418111633204279047022294313820, 9.858068204192527372272313626609

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