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Properties

Label 2-882-63.58-c1-0-16
Degree $2$
Conductor $882$
Sign $0.776 + 0.629i$
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 + (−0.5 + 1.65i)3-s + 4-s + (−0.686 − 1.18i)5-s + (0.5 − 1.65i)6-s − 8-s + (−2.5 − 1.65i)9-s + (0.686 + 1.18i)10-s + (−2.18 + 3.78i)11-s + (−0.5 + 1.65i)12-s + (1 − 1.73i)13-s + (2.31 − 0.543i)15-s + 16-s + (−2.18 − 3.78i)17-s + (2.5 + 1.65i)18-s + (2.5 − 4.33i)19-s + ⋯
L(s)  = 1  − 0.707·2-s + (−0.288 + 0.957i)3-s + 0.5·4-s + (−0.306 − 0.531i)5-s + (0.204 − 0.677i)6-s − 0.353·8-s + (−0.833 − 0.552i)9-s + (0.216 + 0.375i)10-s + (−0.659 + 1.14i)11-s + (−0.144 + 0.478i)12-s + (0.277 − 0.480i)13-s + (0.597 − 0.140i)15-s + 0.250·16-s + (−0.530 − 0.918i)17-s + (0.589 + 0.390i)18-s + (0.573 − 0.993i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.776 + 0.629i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.776 + 0.629i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.776 + 0.629i$
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: \((2,\ 882,\ (\ :1/2),\ 0.776 + 0.629i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.646580 - 0.229174i\)
\(L(\frac12)\) \(\approx\) \(0.646580 - 0.229174i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 - 1.65i)T \)
7 \( 1 \)
good5 \( 1 + (0.686 + 1.18i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (2.18 - 3.78i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1 + 1.73i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (2.18 + 3.78i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.5 + 4.33i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.68 - 6.38i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.37 + 2.37i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 2T + 31T^{2} \)
37 \( 1 + (1 - 1.73i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-5.18 + 8.98i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.55 + 7.89i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + (1.37 + 2.37i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + 7.11T + 59T^{2} \)
61 \( 1 - 14.1T + 61T^{2} \)
67 \( 1 - 15.1T + 67T^{2} \)
71 \( 1 - 10.1T + 71T^{2} \)
73 \( 1 + (2.55 + 4.43i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 - 12.1T + 79T^{2} \)
83 \( 1 + (2.74 + 4.75i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-1.62 + 2.81i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-4.55 - 7.89i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.880728954830915262912487848487, −9.328175873909258669851433132114, −8.610905283260010812174727590985, −7.57880000949199420463330704115, −6.81442951393795254157027431772, −5.35723657218435868243043035227, −4.91540282621348028589242064689, −3.66637222612032362078387306019, −2.43975375425703291348909996971, −0.49726250061093211750543280831, 1.14175170795472192870097505512, 2.50244670026914170383482709836, 3.53199008311516285914807976365, 5.24090715196535114281999016393, 6.27145284296263410050031464904, 6.77312937440100380232909430235, 7.88256576212643953347746870358, 8.283494786215649412132981501325, 9.215766071383522393670521554547, 10.50271012673762189285224488204

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