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Properties

Label 2-693-11.9-c1-0-4
Degree $2$
Conductor $693$
Sign $-0.921 + 0.388i$
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  + (−0.549 + 1.69i)2-s + (−0.938 − 0.681i)4-s + (0.858 + 2.64i)5-s + (0.809 + 0.587i)7-s + (−1.20 + 0.878i)8-s − 4.93·10-s + (−1.14 − 3.11i)11-s + (−1.32 + 4.08i)13-s + (−1.43 + 1.04i)14-s + (−1.53 − 4.73i)16-s + (0.851 + 2.62i)17-s + (−1.56 + 1.13i)19-s + (0.995 − 3.06i)20-s + (5.89 − 0.232i)22-s − 4.37·23-s + ⋯
L(s)  = 1  + (−0.388 + 1.19i)2-s + (−0.469 − 0.340i)4-s + (0.383 + 1.18i)5-s + (0.305 + 0.222i)7-s + (−0.427 + 0.310i)8-s − 1.56·10-s + (−0.346 − 0.938i)11-s + (−0.367 + 1.13i)13-s + (−0.384 + 0.279i)14-s + (−0.384 − 1.18i)16-s + (0.206 + 0.635i)17-s + (−0.359 + 0.261i)19-s + (0.222 − 0.684i)20-s + (1.25 − 0.0495i)22-s − 0.911·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.921 + 0.388i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.921 + 0.388i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(693\)    =    \(3^{2} \cdot 7 \cdot 11\)
Sign: $-0.921 + 0.388i$
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: \((2,\ 693,\ (\ :1/2),\ -0.921 + 0.388i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.209335 - 1.03437i\)
\(L(\frac12)\) \(\approx\) \(0.209335 - 1.03437i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.14 + 3.11i)T \)
good2 \( 1 + (0.549 - 1.69i)T + (-1.61 - 1.17i)T^{2} \)
5 \( 1 + (-0.858 - 2.64i)T + (-4.04 + 2.93i)T^{2} \)
13 \( 1 + (1.32 - 4.08i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (-0.851 - 2.62i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (1.56 - 1.13i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + 4.37T + 23T^{2} \)
29 \( 1 + (-6.98 - 5.07i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (0.0619 - 0.190i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (0.837 + 0.608i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-7.77 + 5.64i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 4.70T + 43T^{2} \)
47 \( 1 + (10.5 - 7.66i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (-1.20 + 3.70i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (6.92 + 5.02i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (0.305 + 0.940i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + 5.41T + 67T^{2} \)
71 \( 1 + (-0.623 - 1.91i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-8.06 - 5.86i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (-1.94 + 5.98i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-0.531 - 1.63i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 - 15.3T + 89T^{2} \)
97 \( 1 + (-3.58 + 11.0i)T + (-78.4 - 57.0i)T^{2} \)
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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

−10.82987550575137351847332126477, −9.996884214673691180903849129730, −8.956849225994281091536681853049, −8.218725778567875352159981054974, −7.40633883620608114704894320586, −6.40368922949805037066493297186, −6.13784190443090454553493728250, −4.89823285178375070822754521596, −3.34012362545872837336299739906, −2.19281909157309866997415209883, 0.61112569032517841190909038833, 1.84059808051410924706269074777, 2.90224228555753674183278225840, 4.38409746298775644446885098699, 5.13838205113702105901761391352, 6.31663346991576764076284008356, 7.69501545349827244292485992588, 8.457459015441587314599834390456, 9.472173828581678450469462193053, 9.989349614340929795246666846118

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