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Properties

Label 2-693-11.9-c1-0-24
Degree $2$
Conductor $693$
Sign $-0.0327 + 0.999i$
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  + (0.123 − 0.381i)2-s + (1.48 + 1.08i)4-s + (−0.712 − 2.19i)5-s + (−0.809 − 0.587i)7-s + (1.24 − 0.904i)8-s − 0.924·10-s + (−3.30 − 0.259i)11-s + (1.38 − 4.27i)13-s + (−0.324 + 0.235i)14-s + (0.945 + 2.91i)16-s + (−2.30 − 7.10i)17-s + (1.08 − 0.791i)19-s + (1.31 − 4.03i)20-s + (−0.508 + 1.22i)22-s + 4.27·23-s + ⋯
L(s)  = 1  + (0.0876 − 0.269i)2-s + (0.743 + 0.540i)4-s + (−0.318 − 0.980i)5-s + (−0.305 − 0.222i)7-s + (0.440 − 0.319i)8-s − 0.292·10-s + (−0.996 − 0.0783i)11-s + (0.384 − 1.18i)13-s + (−0.0866 + 0.0629i)14-s + (0.236 + 0.727i)16-s + (−0.559 − 1.72i)17-s + (0.250 − 0.181i)19-s + (0.293 − 0.901i)20-s + (−0.108 + 0.261i)22-s + 0.890·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0327 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0327 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(693\)    =    \(3^{2} \cdot 7 \cdot 11\)
Sign: $-0.0327 + 0.999i$
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: \((2,\ 693,\ (\ :1/2),\ -0.0327 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.07284 - 1.10853i\)
\(L(\frac12)\) \(\approx\) \(1.07284 - 1.10853i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.30 + 0.259i)T \)
good2 \( 1 + (-0.123 + 0.381i)T + (-1.61 - 1.17i)T^{2} \)
5 \( 1 + (0.712 + 2.19i)T + (-4.04 + 2.93i)T^{2} \)
13 \( 1 + (-1.38 + 4.27i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (2.30 + 7.10i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (-1.08 + 0.791i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 - 4.27T + 23T^{2} \)
29 \( 1 + (-0.910 - 0.661i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (2.50 - 7.71i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (8.95 + 6.50i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (0.939 - 0.682i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 9.95T + 43T^{2} \)
47 \( 1 + (-10.3 + 7.54i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (3.62 - 11.1i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (0.848 + 0.616i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (2.06 + 6.36i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + 2.46T + 67T^{2} \)
71 \( 1 + (2.60 + 8.01i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-8.71 - 6.33i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (-0.837 + 2.57i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-2.82 - 8.68i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + 5.28T + 89T^{2} \)
97 \( 1 + (-1.77 + 5.46i)T + (-78.4 - 57.0i)T^{2} \)
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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

−10.68392900971115899368513210133, −9.256877687865278920817400739829, −8.523670600288109299721972317951, −7.54012307977364599258972446947, −7.00151766137716751631536910350, −5.53563421020220262907342394329, −4.76911156401650936234788017097, −3.42862842324746636139677520977, −2.58991614284172155713281783549, −0.77691144340622189717707533824, 1.87074170737323237109439750947, 2.92970310906882582394598246129, 4.18218274659993237769432013232, 5.54420620864868527854493710220, 6.36387183706338104969369685180, 7.01193129524483501203024429321, 7.84927942778620955860497000461, 8.944325132748394425771785818729, 10.06699312977978504809883995178, 10.83440585952212138726959914701

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