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Properties

Label 2-882-63.58-c1-0-10
Degree $2$
Conductor $882$
Sign $0.631 - 0.775i$
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 + (−1.09 − 1.34i)3-s + 4-s + (0.880 + 1.52i)5-s + (−1.09 − 1.34i)6-s + 8-s + (−0.619 + 2.93i)9-s + (0.880 + 1.52i)10-s + (−3.06 + 5.30i)11-s + (−1.09 − 1.34i)12-s + (0.380 − 0.658i)13-s + (1.09 − 2.84i)15-s + 16-s + (3.42 + 5.92i)17-s + (−0.619 + 2.93i)18-s + (−0.971 + 1.68i)19-s + ⋯
L(s)  = 1  + 0.707·2-s + (−0.629 − 0.776i)3-s + 0.5·4-s + (0.393 + 0.681i)5-s + (−0.445 − 0.549i)6-s + 0.353·8-s + (−0.206 + 0.978i)9-s + (0.278 + 0.482i)10-s + (−0.923 + 1.59i)11-s + (−0.314 − 0.388i)12-s + (0.105 − 0.182i)13-s + (0.281 − 0.735i)15-s + 0.250·16-s + (0.829 + 1.43i)17-s + (−0.146 + 0.691i)18-s + (−0.222 + 0.385i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.631 - 0.775i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.631 - 0.775i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.631 - 0.775i$
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: \((2,\ 882,\ (\ :1/2),\ 0.631 - 0.775i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.66705 + 0.791919i\)
\(L(\frac12)\) \(\approx\) \(1.66705 + 0.791919i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.09 + 1.34i)T \)
7 \( 1 \)
good5 \( 1 + (-0.880 - 1.52i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (3.06 - 5.30i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.380 + 0.658i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-3.42 - 5.92i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.971 - 1.68i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.210 - 0.364i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.732 - 1.26i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 7.70T + 31T^{2} \)
37 \( 1 + (-1.44 + 2.49i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-3.47 + 6.01i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4.33 - 7.49i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 1.66T + 47T^{2} \)
53 \( 1 + (0.112 + 0.195i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + 1.98T + 59T^{2} \)
61 \( 1 - 10.3T + 61T^{2} \)
67 \( 1 - 6.78T + 67T^{2} \)
71 \( 1 - 10.7T + 71T^{2} \)
73 \( 1 + (0.153 + 0.265i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + 13.4T + 79T^{2} \)
83 \( 1 + (-1.56 - 2.70i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (1.30 - 2.25i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-1.81 - 3.14i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.53645194965738914082495641449, −9.781007950192324834849425218982, −8.163190759306060888768857796229, −7.46645571359859157912370578108, −6.74071279120476809342871655498, −5.86423557268263855185664943656, −5.20346815821490426441133462994, −4.01441416701342527771533067883, −2.57626016281420032575854290572, −1.73120059155706083279424728582, 0.76624280172878468688619331844, 2.79002220357769867237199400985, 3.72293680967607632182675396621, 4.98480657840510896700648108856, 5.38660549627724443870281180386, 6.13333734390254334734356235413, 7.30599286239820552482913496138, 8.488456106436409215218583549599, 9.283706940155011155946098291418, 10.11165688024242958518974384559

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