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Properties

Label 2-693-11.5-c1-0-29
Degree $2$
Conductor $693$
Sign $0.162 - 0.986i$
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  + (−0.804 − 2.47i)2-s + (−3.86 + 2.80i)4-s + (1.33 − 4.10i)5-s + (−0.809 + 0.587i)7-s + (5.85 + 4.25i)8-s − 11.2·10-s + (−3.29 − 0.388i)11-s + (−0.761 − 2.34i)13-s + (2.10 + 1.53i)14-s + (2.87 − 8.83i)16-s + (0.366 − 1.12i)17-s + (2.77 + 2.01i)19-s + (6.37 + 19.6i)20-s + (1.68 + 8.46i)22-s − 8.32·23-s + ⋯
L(s)  = 1  + (−0.568 − 1.75i)2-s + (−1.93 + 1.40i)4-s + (0.596 − 1.83i)5-s + (−0.305 + 0.222i)7-s + (2.07 + 1.50i)8-s − 3.55·10-s + (−0.993 − 0.117i)11-s + (−0.211 − 0.650i)13-s + (0.563 + 0.409i)14-s + (0.717 − 2.20i)16-s + (0.0889 − 0.273i)17-s + (0.636 + 0.462i)19-s + (1.42 + 4.39i)20-s + (0.359 + 1.80i)22-s − 1.73·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.162 - 0.986i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.162 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(693\)    =    \(3^{2} \cdot 7 \cdot 11\)
Sign: $0.162 - 0.986i$
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: \((2,\ 693,\ (\ :1/2),\ 0.162 - 0.986i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.400402 + 0.339875i\)
\(L(\frac12)\) \(\approx\) \(0.400402 + 0.339875i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.29 + 0.388i)T \)
good2 \( 1 + (0.804 + 2.47i)T + (-1.61 + 1.17i)T^{2} \)
5 \( 1 + (-1.33 + 4.10i)T + (-4.04 - 2.93i)T^{2} \)
13 \( 1 + (0.761 + 2.34i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (-0.366 + 1.12i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (-2.77 - 2.01i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + 8.32T + 23T^{2} \)
29 \( 1 + (-1.17 + 0.854i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-0.923 - 2.84i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (5.13 - 3.72i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (4.98 + 3.62i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 3.15T + 43T^{2} \)
47 \( 1 + (-7.06 - 5.13i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (-2.06 - 6.35i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-9.50 + 6.90i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-2.40 + 7.38i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 - 3.11T + 67T^{2} \)
71 \( 1 + (1.69 - 5.21i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (7.00 - 5.09i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (2.73 + 8.40i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-2.98 + 9.19i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + 0.985T + 89T^{2} \)
97 \( 1 + (-2.15 - 6.63i)T + (-78.4 + 57.0i)T^{2} \)
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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

−10.02395990276030454521391584190, −9.167504950614110180067426608236, −8.417627141611570870427321158400, −7.86102373033381426968367789971, −5.69910656821384008863461179916, −5.00902397512239387964852459593, −3.94477714074376039698619910351, −2.65296262436041409388439753861, −1.57693026978513272227815426513, −0.33231279945503582123843986936, 2.37031260874182308334595352223, 3.88810577965106696684183705453, 5.38087729776146021566162974622, 6.07369720275490685430863238422, 6.91235448576838574379588297002, 7.34316844907814277109562735441, 8.225879611307792751117408309125, 9.425537988183697846490280736348, 10.11558013310290779580617931674, 10.52487483689169336984072049440

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