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Properties

Label 2-828-23.16-c1-0-9
Degree $2$
Conductor $828$
Sign $-0.977 - 0.210i$
Analytic cond. $6.61161$
Root an. cond. $2.57130$
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  + (−1.11 − 2.44i)5-s + (0.161 − 0.0474i)7-s + (−3.48 − 4.02i)11-s + (−3.68 − 1.08i)13-s + (−0.947 + 6.58i)17-s + (0.980 + 6.82i)19-s + (−2.24 + 4.23i)23-s + (−1.45 + 1.68i)25-s + (−0.0811 + 0.564i)29-s + (−5.85 − 3.76i)31-s + (−0.296 − 0.342i)35-s + (3.67 − 8.05i)37-s + (−2.02 − 4.42i)41-s + (−5.68 + 3.65i)43-s + 2.26·47-s + ⋯
L(s)  = 1  + (−0.499 − 1.09i)5-s + (0.0610 − 0.0179i)7-s + (−1.05 − 1.21i)11-s + (−1.02 − 0.300i)13-s + (−0.229 + 1.59i)17-s + (0.225 + 1.56i)19-s + (−0.467 + 0.884i)23-s + (−0.291 + 0.336i)25-s + (−0.0150 + 0.104i)29-s + (−1.05 − 0.675i)31-s + (−0.0501 − 0.0578i)35-s + (0.604 − 1.32i)37-s + (−0.315 − 0.691i)41-s + (−0.866 + 0.557i)43-s + 0.330·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 828 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.977 - 0.210i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 828 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.977 - 0.210i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(828\)    =    \(2^{2} \cdot 3^{2} \cdot 23\)
Sign: $-0.977 - 0.210i$
Analytic conductor: \(6.61161\)
Root analytic conductor: \(2.57130\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{828} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 828,\ (\ :1/2),\ -0.977 - 0.210i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0240096 + 0.225105i\)
\(L(\frac12)\) \(\approx\) \(0.0240096 + 0.225105i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
23 \( 1 + (2.24 - 4.23i)T \)
good5 \( 1 + (1.11 + 2.44i)T + (-3.27 + 3.77i)T^{2} \)
7 \( 1 + (-0.161 + 0.0474i)T + (5.88 - 3.78i)T^{2} \)
11 \( 1 + (3.48 + 4.02i)T + (-1.56 + 10.8i)T^{2} \)
13 \( 1 + (3.68 + 1.08i)T + (10.9 + 7.02i)T^{2} \)
17 \( 1 + (0.947 - 6.58i)T + (-16.3 - 4.78i)T^{2} \)
19 \( 1 + (-0.980 - 6.82i)T + (-18.2 + 5.35i)T^{2} \)
29 \( 1 + (0.0811 - 0.564i)T + (-27.8 - 8.17i)T^{2} \)
31 \( 1 + (5.85 + 3.76i)T + (12.8 + 28.1i)T^{2} \)
37 \( 1 + (-3.67 + 8.05i)T + (-24.2 - 27.9i)T^{2} \)
41 \( 1 + (2.02 + 4.42i)T + (-26.8 + 30.9i)T^{2} \)
43 \( 1 + (5.68 - 3.65i)T + (17.8 - 39.1i)T^{2} \)
47 \( 1 - 2.26T + 47T^{2} \)
53 \( 1 + (-4.67 + 1.37i)T + (44.5 - 28.6i)T^{2} \)
59 \( 1 + (-2.18 - 0.642i)T + (49.6 + 31.8i)T^{2} \)
61 \( 1 + (11.7 + 7.54i)T + (25.3 + 55.4i)T^{2} \)
67 \( 1 + (-1.05 + 1.21i)T + (-9.53 - 66.3i)T^{2} \)
71 \( 1 + (6.21 - 7.17i)T + (-10.1 - 70.2i)T^{2} \)
73 \( 1 + (0.545 + 3.79i)T + (-70.0 + 20.5i)T^{2} \)
79 \( 1 + (2.51 + 0.739i)T + (66.4 + 42.7i)T^{2} \)
83 \( 1 + (-3.72 + 8.15i)T + (-54.3 - 62.7i)T^{2} \)
89 \( 1 + (5.28 - 3.39i)T + (36.9 - 80.9i)T^{2} \)
97 \( 1 + (5.51 + 12.0i)T + (-63.5 + 73.3i)T^{2} \)
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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

−9.793019748599666220386345154514, −8.741245724232136883268153366305, −8.043957795256167982088987314559, −7.59203122951330905687785828744, −5.91033357732540345519489967616, −5.46513470695241474990822215901, −4.28051634006057239595738775276, −3.36751999325769803024969342777, −1.77256813998380322887384512733, −0.10410062398160712506842907251, 2.38960098535853493583389951700, 2.97890713645001880431405667760, 4.59280789588477190371165949829, 5.09017373133874014652804039829, 6.80702269791790549154104061045, 7.10054336110958690106848836596, 7.85650205597540957595579960570, 9.129369063283060218560670849302, 9.932015989459517632198854245427, 10.61128196762559054881214915629

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