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Properties

Label 2-693-11.9-c1-0-28
Degree $2$
Conductor $693$
Sign $-0.324 - 0.945i$
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  + (0.751 − 2.31i)2-s + (−3.16 − 2.29i)4-s + (−0.388 − 1.19i)5-s + (−0.809 − 0.587i)7-s + (−3.74 + 2.72i)8-s − 3.05·10-s + (−3.18 + 0.930i)11-s + (−0.982 + 3.02i)13-s + (−1.96 + 1.42i)14-s + (1.06 + 3.28i)16-s + (−1.83 − 5.63i)17-s + (−2.31 + 1.68i)19-s + (−1.51 + 4.67i)20-s + (−0.239 + 8.05i)22-s − 6.76·23-s + ⋯
L(s)  = 1  + (0.531 − 1.63i)2-s + (−1.58 − 1.14i)4-s + (−0.173 − 0.535i)5-s + (−0.305 − 0.222i)7-s + (−1.32 + 0.963i)8-s − 0.967·10-s + (−0.959 + 0.280i)11-s + (−0.272 + 0.838i)13-s + (−0.525 + 0.381i)14-s + (0.266 + 0.820i)16-s + (−0.444 − 1.36i)17-s + (−0.531 + 0.385i)19-s + (−0.339 + 1.04i)20-s + (−0.0509 + 1.71i)22-s − 1.41·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.324 - 0.945i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.324 - 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(693\)    =    \(3^{2} \cdot 7 \cdot 11\)
Sign: $-0.324 - 0.945i$
Analytic conductor: \(5.53363\)
Root analytic conductor: \(2.35236\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{693} (64, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 693,\ (\ :1/2),\ -0.324 - 0.945i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.534520 + 0.748284i\)
\(L(\frac12)\) \(\approx\) \(0.534520 + 0.748284i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.809 + 0.587i)T \)
11 \( 1 + (3.18 - 0.930i)T \)
good2 \( 1 + (-0.751 + 2.31i)T + (-1.61 - 1.17i)T^{2} \)
5 \( 1 + (0.388 + 1.19i)T + (-4.04 + 2.93i)T^{2} \)
13 \( 1 + (0.982 - 3.02i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (1.83 + 5.63i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (2.31 - 1.68i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + 6.76T + 23T^{2} \)
29 \( 1 + (-3.63 - 2.64i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-3.00 + 9.25i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-4.41 - 3.20i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (0.254 - 0.184i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 0.132T + 43T^{2} \)
47 \( 1 + (-7.58 + 5.51i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (1.34 - 4.13i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (5.62 + 4.08i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (0.757 + 2.33i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + 9.41T + 67T^{2} \)
71 \( 1 + (-0.0360 - 0.110i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (0.497 + 0.361i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (-2.63 + 8.11i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (0.293 + 0.904i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + 10.0T + 89T^{2} \)
97 \( 1 + (-5.43 + 16.7i)T + (-78.4 - 57.0i)T^{2} \)
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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

−10.00554376386670620813237199781, −9.450184932411985061885190496545, −8.423241680122501781237368056955, −7.28497692621086431629723150434, −5.94196179156491612939942667361, −4.64990328682012107143723530385, −4.34335498375454096880249710011, −2.92567238284116980205291968641, −2.04155489971544831492316267957, −0.38839322476315926135571284020, 2.72576157625927281509584688199, 3.93182735688967248531973835783, 4.97431384230307203673362582427, 5.93933381231595740257829839784, 6.48384602386003664851499984466, 7.52563001330848690859611087588, 8.160277054587208293936748642758, 8.871857731102569320696073928246, 10.26534833768901916956585122659, 10.81167214514408717727884641280

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