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Properties

Label 2-693-9.4-c1-0-34
Degree $2$
Conductor $693$
Sign $0.690 + 0.723i$
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  + (−0.313 − 0.543i)2-s + (−0.395 + 1.68i)3-s + (0.803 − 1.39i)4-s + (−0.166 + 0.288i)5-s + (1.03 − 0.314i)6-s + (0.5 + 0.866i)7-s − 2.26·8-s + (−2.68 − 1.33i)9-s + 0.208·10-s + (−0.5 − 0.866i)11-s + (2.02 + 1.90i)12-s + (3.15 − 5.47i)13-s + (0.313 − 0.543i)14-s + (−0.420 − 0.394i)15-s + (−0.897 − 1.55i)16-s + 2.34·17-s + ⋯
L(s)  = 1  + (−0.221 − 0.384i)2-s + (−0.228 + 0.973i)3-s + (0.401 − 0.695i)4-s + (−0.0744 + 0.128i)5-s + (0.424 − 0.128i)6-s + (0.188 + 0.327i)7-s − 0.799·8-s + (−0.895 − 0.444i)9-s + 0.0660·10-s + (−0.150 − 0.261i)11-s + (0.585 + 0.549i)12-s + (0.876 − 1.51i)13-s + (0.0838 − 0.145i)14-s + (−0.108 − 0.101i)15-s + (−0.224 − 0.388i)16-s + 0.568·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.690 + 0.723i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.690 + 0.723i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(693\)    =    \(3^{2} \cdot 7 \cdot 11\)
Sign: $0.690 + 0.723i$
Analytic conductor: \(5.53363\)
Root analytic conductor: \(2.35236\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{693} (463, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 693,\ (\ :1/2),\ 0.690 + 0.723i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.19718 - 0.512680i\)
\(L(\frac12)\) \(\approx\) \(1.19718 - 0.512680i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.395 - 1.68i)T \)
7 \( 1 + (-0.5 - 0.866i)T \)
11 \( 1 + (0.5 + 0.866i)T \)
good2 \( 1 + (0.313 + 0.543i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (0.166 - 0.288i)T + (-2.5 - 4.33i)T^{2} \)
13 \( 1 + (-3.15 + 5.47i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 2.34T + 17T^{2} \)
19 \( 1 - 2.12T + 19T^{2} \)
23 \( 1 + (-1.69 + 2.93i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.13 - 1.96i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3.58 + 6.20i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 0.952T + 37T^{2} \)
41 \( 1 + (1.75 - 3.03i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.05 + 3.56i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (2.17 + 3.76i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 5.73T + 53T^{2} \)
59 \( 1 + (-1.11 + 1.92i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.29 - 3.97i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.70 + 8.14i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 11.2T + 71T^{2} \)
73 \( 1 - 8.83T + 73T^{2} \)
79 \( 1 + (0.819 + 1.41i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (5.05 + 8.74i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 13.3T + 89T^{2} \)
97 \( 1 + (5.53 + 9.58i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.39157465819943227732476848542, −9.769665217040712301065934566716, −8.824911669892293303399536375046, −8.013714579795362371296493050083, −6.56100555156293554453153939455, −5.62484131240088257394712939916, −5.13941952747314680325428134093, −3.54621710467545481865693230777, −2.73424798719910161404405467203, −0.854520649271376930725990920575, 1.38107258486660314511348462629, 2.73737904901597907349506500635, 4.00762752137386192173366176330, 5.37700200836824373587565385087, 6.63379114259225592233000366685, 6.88104041551548730830763989024, 7.977014228939661910708543814743, 8.477661546697628108437715090978, 9.483855829046353193259105729336, 10.83900617810170355754074342928

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