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Properties

Label 2-693-11.5-c1-0-2
Degree $2$
Conductor $693$
Sign $0.998 - 0.0475i$
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  + (−0.690 − 2.12i)2-s + (−2.42 + 1.76i)4-s + (0.190 − 0.587i)5-s + (−0.809 + 0.587i)7-s + (1.80 + 1.31i)8-s − 1.38·10-s + (0.309 + 3.30i)11-s + (1 + 3.07i)13-s + (1.80 + 1.31i)14-s + (−0.309 + 0.951i)16-s + (−1.5 + 4.61i)17-s + (−2.30 − 1.67i)19-s + (0.572 + 1.76i)20-s + (6.80 − 2.93i)22-s − 4.38·23-s + ⋯
L(s)  = 1  + (−0.488 − 1.50i)2-s + (−1.21 + 0.881i)4-s + (0.0854 − 0.262i)5-s + (−0.305 + 0.222i)7-s + (0.639 + 0.464i)8-s − 0.437·10-s + (0.0931 + 0.995i)11-s + (0.277 + 0.853i)13-s + (0.483 + 0.351i)14-s + (−0.0772 + 0.237i)16-s + (−0.363 + 1.11i)17-s + (−0.529 − 0.384i)19-s + (0.128 + 0.394i)20-s + (1.45 − 0.626i)22-s − 0.913·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0475i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 - 0.0475i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(693\)    =    \(3^{2} \cdot 7 \cdot 11\)
Sign: $0.998 - 0.0475i$
Analytic conductor: \(5.53363\)
Root analytic conductor: \(2.35236\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{693} (379, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 693,\ (\ :1/2),\ 0.998 - 0.0475i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.696805 + 0.0165689i\)
\(L(\frac12)\) \(\approx\) \(0.696805 + 0.0165689i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.309 - 3.30i)T \)
good2 \( 1 + (0.690 + 2.12i)T + (-1.61 + 1.17i)T^{2} \)
5 \( 1 + (-0.190 + 0.587i)T + (-4.04 - 2.93i)T^{2} \)
13 \( 1 + (-1 - 3.07i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (1.5 - 4.61i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (2.30 + 1.67i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + 4.38T + 23T^{2} \)
29 \( 1 + (4.85 - 3.52i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (0.954 + 2.93i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (3.73 - 2.71i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (-5.97 - 4.33i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 9.70T + 43T^{2} \)
47 \( 1 + (-3.61 - 2.62i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (2.09 + 6.43i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-2.61 + 1.90i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + 67T^{2} \)
71 \( 1 + (-1.52 + 4.70i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (11.0 - 8.05i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (3.09 + 9.51i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (5.09 - 15.6i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + 5.61T + 89T^{2} \)
97 \( 1 + (1.85 + 5.70i)T + (-78.4 + 57.0i)T^{2} \)
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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

−10.57047295787860151999541236594, −9.612484294952288660251096755733, −9.129538251150867014033146070278, −8.317154493431068092074149244723, −7.02946013813497735730244987672, −5.99177639184461849666941453355, −4.49749686978425586062707623859, −3.77583285364392106280796744713, −2.39201394884247906088713761290, −1.53523579922300743074563566764, 0.43763971164208055440163190000, 2.79409494241422522582195023145, 4.17139758756295600897947975613, 5.57114258060562523641974237498, 6.03209973110451030084505500144, 7.02737752621503957268480407873, 7.72283418653168875028480814210, 8.613317385299440839646470613237, 9.237504624507935888773196621679, 10.27906266221992906137338759779

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