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Properties

Label 2-792-88.19-c1-0-13
Degree $2$
Conductor $792$
Sign $-0.719 - 0.694i$
Analytic cond. $6.32415$
Root an. cond. $2.51478$
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  + (−1.03 + 0.965i)2-s + (0.136 − 1.99i)4-s + (−0.409 + 0.563i)5-s + (0.510 − 1.57i)7-s + (1.78 + 2.19i)8-s + (−0.120 − 0.977i)10-s + (−2.08 − 2.58i)11-s + (−4.92 + 3.57i)13-s + (0.988 + 2.11i)14-s + (−3.96 − 0.544i)16-s + (−2.23 + 3.07i)17-s + (5.66 − 1.84i)19-s + (1.06 + 0.893i)20-s + (4.64 + 0.655i)22-s + 8.64i·23-s + ⋯
L(s)  = 1  + (−0.730 + 0.682i)2-s + (0.0682 − 0.997i)4-s + (−0.183 + 0.251i)5-s + (0.192 − 0.593i)7-s + (0.631 + 0.775i)8-s + (−0.0381 − 0.309i)10-s + (−0.628 − 0.778i)11-s + (−1.36 + 0.991i)13-s + (0.264 + 0.565i)14-s + (−0.990 − 0.136i)16-s + (−0.541 + 0.745i)17-s + (1.29 − 0.422i)19-s + (0.238 + 0.199i)20-s + (0.990 + 0.139i)22-s + 1.80i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.719 - 0.694i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 792 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.719 - 0.694i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(792\)    =    \(2^{3} \cdot 3^{2} \cdot 11\)
Sign: $-0.719 - 0.694i$
Analytic conductor: \(6.32415\)
Root analytic conductor: \(2.51478\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{792} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 792,\ (\ :1/2),\ -0.719 - 0.694i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.226166 + 0.560209i\)
\(L(\frac12)\) \(\approx\) \(0.226166 + 0.560209i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.03 - 0.965i)T \)
3 \( 1 \)
11 \( 1 + (2.08 + 2.58i)T \)
good5 \( 1 + (0.409 - 0.563i)T + (-1.54 - 4.75i)T^{2} \)
7 \( 1 + (-0.510 + 1.57i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (4.92 - 3.57i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (2.23 - 3.07i)T + (-5.25 - 16.1i)T^{2} \)
19 \( 1 + (-5.66 + 1.84i)T + (15.3 - 11.1i)T^{2} \)
23 \( 1 - 8.64iT - 23T^{2} \)
29 \( 1 + (-1.33 + 4.10i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-4.09 - 5.64i)T + (-9.57 + 29.4i)T^{2} \)
37 \( 1 + (2.02 + 0.657i)T + (29.9 + 21.7i)T^{2} \)
41 \( 1 + (0.736 - 0.239i)T + (33.1 - 24.0i)T^{2} \)
43 \( 1 - 10.1iT - 43T^{2} \)
47 \( 1 + (5.29 - 1.72i)T + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (2.57 + 3.54i)T + (-16.3 + 50.4i)T^{2} \)
59 \( 1 + (3.78 - 11.6i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (0.239 + 0.174i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 10.7T + 67T^{2} \)
71 \( 1 + (-1.14 + 1.56i)T + (-21.9 - 67.5i)T^{2} \)
73 \( 1 + (6.90 + 2.24i)T + (59.0 + 42.9i)T^{2} \)
79 \( 1 + (-2.76 + 2.01i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-1.90 + 2.62i)T + (-25.6 - 78.9i)T^{2} \)
89 \( 1 - 6.63T + 89T^{2} \)
97 \( 1 + (3.48 - 2.53i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.43621909679853444868450745122, −9.634467100758747892009990165116, −8.949068916599646206887483968412, −7.75680342079023659557636035530, −7.41052433437453122379477943440, −6.46765235880777632770198615470, −5.37934893973195575593942760828, −4.53656819552080920764399568080, −3.00684614760164191482642628019, −1.43390129325391522828863983662, 0.39205902664156144068824014770, 2.25552565386119028176137357241, 2.93584492786320908386028901732, 4.53904754793631770439314358697, 5.21033841292295499888923330651, 6.80054112959906060194264783327, 7.65051799981636130473028351441, 8.306454423747472110445570660011, 9.212988291650599331721481176954, 10.07319238030353682807363034523

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