Properties

Label 2-693-11.4-c1-0-3
Degree $2$
Conductor $693$
Sign $0.970 - 0.242i$
Analytic cond. $5.53363$
Root an. cond. $2.35236$
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  + (−1.93 − 1.40i)2-s + (1.14 + 3.53i)4-s + (−2.09 + 1.52i)5-s + (−0.309 − 0.951i)7-s + (1.26 − 3.89i)8-s + 6.19·10-s + (2.19 − 2.48i)11-s + (−5.48 − 3.98i)13-s + (−0.738 + 2.27i)14-s + (−1.91 + 1.38i)16-s + (−2.54 + 1.84i)17-s + (−0.323 + 0.994i)19-s + (−7.78 − 5.65i)20-s + (−7.73 + 1.73i)22-s + 6.52·23-s + ⋯
L(s)  = 1  + (−1.36 − 0.993i)2-s + (0.573 + 1.76i)4-s + (−0.937 + 0.680i)5-s + (−0.116 − 0.359i)7-s + (0.447 − 1.37i)8-s + 1.95·10-s + (0.660 − 0.750i)11-s + (−1.52 − 1.10i)13-s + (−0.197 + 0.607i)14-s + (−0.477 + 0.347i)16-s + (−0.617 + 0.448i)17-s + (−0.0741 + 0.228i)19-s + (−1.73 − 1.26i)20-s + (−1.64 + 0.370i)22-s + 1.36·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 - 0.242i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.970 - 0.242i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(693\)    =    \(3^{2} \cdot 7 \cdot 11\)
Sign: $0.970 - 0.242i$
Analytic conductor: \(5.53363\)
Root analytic conductor: \(2.35236\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{693} (631, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 693,\ (\ :1/2),\ 0.970 - 0.242i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.420075 + 0.0516384i\)
\(L(\frac12)\) \(\approx\) \(0.420075 + 0.0516384i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 + (0.309 + 0.951i)T \)
11 \( 1 + (-2.19 + 2.48i)T \)
good2 \( 1 + (1.93 + 1.40i)T + (0.618 + 1.90i)T^{2} \)
5 \( 1 + (2.09 - 1.52i)T + (1.54 - 4.75i)T^{2} \)
13 \( 1 + (5.48 + 3.98i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (2.54 - 1.84i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (0.323 - 0.994i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 - 6.52T + 23T^{2} \)
29 \( 1 + (-0.187 - 0.577i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-7.14 - 5.19i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-2.76 - 8.50i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (2.68 - 8.27i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 4.48T + 43T^{2} \)
47 \( 1 + (1.00 - 3.10i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-1.53 - 1.11i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-0.0537 - 0.165i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (6.58 - 4.78i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + 7.33T + 67T^{2} \)
71 \( 1 + (-2.18 + 1.59i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-0.182 - 0.561i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-7.93 - 5.76i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-7.07 + 5.14i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 - 11.6T + 89T^{2} \)
97 \( 1 + (9.58 + 6.96i)T + (29.9 + 92.2i)T^{2} \)
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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

−10.51771587934842204657482151315, −9.820950215381299856456769845805, −8.849079067543409249357565133513, −8.034073336245644924312864937841, −7.40842592714879457713587900793, −6.47829225724700895919564061923, −4.72766586667046250502516333514, −3.33034771062405433616187636670, −2.79896820090622258414380207715, −0.989998616129169016570699019480, 0.45327886048870740603271092981, 2.18180200513117894507987618477, 4.25350687694801348530327922462, 5.04362459710802322135527322786, 6.45558686671398586530251327611, 7.18444111830527103062675753172, 7.73020096477504756141792021124, 8.926883734487521120206394868030, 9.155967469904121103750693325254, 9.963354510125776690572868302988

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