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Properties

Label 2-693-11.3-c1-0-10
Degree $2$
Conductor $693$
Sign $0.990 - 0.141i$
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  + (0.705 − 0.512i)2-s + (−0.383 + 1.17i)4-s + (−3.28 − 2.38i)5-s + (−0.309 + 0.951i)7-s + (0.872 + 2.68i)8-s − 3.54·10-s + (2.96 + 1.48i)11-s + (4.92 − 3.57i)13-s + (0.269 + 0.828i)14-s + (−0.0153 − 0.0111i)16-s + (1.37 + 1.00i)17-s + (1.68 + 5.19i)19-s + (4.07 − 2.96i)20-s + (2.85 − 0.475i)22-s + 6.39·23-s + ⋯
L(s)  = 1  + (0.498 − 0.362i)2-s + (−0.191 + 0.589i)4-s + (−1.47 − 1.06i)5-s + (−0.116 + 0.359i)7-s + (0.308 + 0.949i)8-s − 1.12·10-s + (0.894 + 0.446i)11-s + (1.36 − 0.992i)13-s + (0.0719 + 0.221i)14-s + (−0.00383 − 0.00278i)16-s + (0.333 + 0.242i)17-s + (0.386 + 1.19i)19-s + (0.911 − 0.662i)20-s + (0.607 − 0.101i)22-s + 1.33·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 - 0.141i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.990 - 0.141i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(693\)    =    \(3^{2} \cdot 7 \cdot 11\)
Sign: $0.990 - 0.141i$
Analytic conductor: \(5.53363\)
Root analytic conductor: \(2.35236\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{693} (190, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 693,\ (\ :1/2),\ 0.990 - 0.141i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.55428 + 0.110154i\)
\(L(\frac12)\) \(\approx\) \(1.55428 + 0.110154i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.96 - 1.48i)T \)
good2 \( 1 + (-0.705 + 0.512i)T + (0.618 - 1.90i)T^{2} \)
5 \( 1 + (3.28 + 2.38i)T + (1.54 + 4.75i)T^{2} \)
13 \( 1 + (-4.92 + 3.57i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-1.37 - 1.00i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-1.68 - 5.19i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 - 6.39T + 23T^{2} \)
29 \( 1 + (1.46 - 4.51i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-3.52 + 2.56i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-0.753 + 2.32i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-0.992 - 3.05i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 0.127T + 43T^{2} \)
47 \( 1 + (2.61 + 8.05i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (3.81 - 2.77i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (1.43 - 4.40i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (4.29 + 3.12i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 - 14.0T + 67T^{2} \)
71 \( 1 + (-1.79 - 1.30i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (1.72 - 5.30i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (5.07 - 3.68i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (0.103 + 0.0750i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 - 8.12T + 89T^{2} \)
97 \( 1 + (-2.10 + 1.52i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.87705458610398318866569146694, −9.408284016430483483091457598864, −8.481895971373222628221855975262, −8.169174571628058351324785333569, −7.19125162090193642122439055811, −5.69694707782005683333884354286, −4.75397326623092753833373253226, −3.79853929330201880654972173294, −3.31673409689377685013773682362, −1.22682436896280497847880895910, 0.925919903765346863483833025066, 3.20739299895461119714706585920, 3.94450686624378062698555202614, 4.76643648997716213173344445242, 6.33742290869157841645976343067, 6.69864501504038925707230354036, 7.53825235733669113386760866884, 8.715819015949048905335335862337, 9.538479437204098899191198872922, 10.79399101331340657681767233852

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