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Properties

Label 2-693-11.9-c1-0-16
Degree $2$
Conductor $693$
Sign $0.988 + 0.154i$
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  + (0.240 − 0.739i)2-s + (1.12 + 0.820i)4-s + (0.0687 + 0.211i)5-s + (0.809 + 0.587i)7-s + (2.13 − 1.55i)8-s + 0.173·10-s + (−0.660 + 3.25i)11-s + (2.01 − 6.20i)13-s + (0.628 − 0.456i)14-s + (0.228 + 0.702i)16-s + (1.33 + 4.12i)17-s + (−2.35 + 1.71i)19-s + (−0.0959 + 0.295i)20-s + (2.24 + 1.26i)22-s + 3.89·23-s + ⋯
L(s)  = 1  + (0.169 − 0.522i)2-s + (0.564 + 0.410i)4-s + (0.0307 + 0.0946i)5-s + (0.305 + 0.222i)7-s + (0.755 − 0.548i)8-s + 0.0547·10-s + (−0.199 + 0.979i)11-s + (0.559 − 1.72i)13-s + (0.168 − 0.122i)14-s + (0.0570 + 0.175i)16-s + (0.324 + 0.999i)17-s + (−0.541 + 0.393i)19-s + (−0.0214 + 0.0660i)20-s + (0.478 + 0.270i)22-s + 0.812·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.988 + 0.154i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.988 + 0.154i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(693\)    =    \(3^{2} \cdot 7 \cdot 11\)
Sign: $0.988 + 0.154i$
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: \((2,\ 693,\ (\ :1/2),\ 0.988 + 0.154i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.09747 - 0.162604i\)
\(L(\frac12)\) \(\approx\) \(2.09747 - 0.162604i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.660 - 3.25i)T \)
good2 \( 1 + (-0.240 + 0.739i)T + (-1.61 - 1.17i)T^{2} \)
5 \( 1 + (-0.0687 - 0.211i)T + (-4.04 + 2.93i)T^{2} \)
13 \( 1 + (-2.01 + 6.20i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (-1.33 - 4.12i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (2.35 - 1.71i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 - 3.89T + 23T^{2} \)
29 \( 1 + (3.05 + 2.21i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (2.12 - 6.55i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-4.57 - 3.32i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-1.08 + 0.786i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 4.70T + 43T^{2} \)
47 \( 1 + (-4.89 + 3.55i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (-0.530 + 1.63i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (7.71 + 5.60i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (2.97 + 9.15i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 - 1.27T + 67T^{2} \)
71 \( 1 + (-2.87 - 8.85i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (4.52 + 3.28i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (1.39 - 4.30i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (3.48 + 10.7i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + 7.92T + 89T^{2} \)
97 \( 1 + (2.79 - 8.61i)T + (-78.4 - 57.0i)T^{2} \)
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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

−10.59104239120931500833541681774, −9.950782830431930530718729412444, −8.510798542540591906675651541588, −7.917942075132932308363198387578, −6.98642076664693359114019201421, −5.96746981727650983435628900637, −4.85317115330333322834557118723, −3.66204220486828299916698981816, −2.73155754064788278777851565508, −1.50771538840841610261496673702, 1.29684674837659620787922586149, 2.68108531054094411692549867000, 4.18241997994324040707537979975, 5.17984914935067249001350019904, 6.08412878180283748736728449383, 6.93257827792753498431590288643, 7.62114734558403423325960827231, 8.812023596885215048020758354820, 9.431212449850704662453907628480, 10.86238254773225349936347666958

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