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Properties

Label 2-693-11.4-c1-0-7
Degree $2$
Conductor $693$
Sign $0.828 + 0.560i$
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  + (−2.16 − 1.57i)2-s + (1.60 + 4.93i)4-s + (0.946 − 0.687i)5-s + (0.309 + 0.951i)7-s + (2.64 − 8.12i)8-s − 3.13·10-s + (3.31 − 0.0332i)11-s + (1.36 + 0.989i)13-s + (0.828 − 2.54i)14-s + (−10.1 + 7.36i)16-s + (−4.52 + 3.28i)17-s + (1.34 − 4.14i)19-s + (4.91 + 3.56i)20-s + (−7.24 − 5.15i)22-s + 0.119·23-s + ⋯
L(s)  = 1  + (−1.53 − 1.11i)2-s + (0.801 + 2.46i)4-s + (0.423 − 0.307i)5-s + (0.116 + 0.359i)7-s + (0.933 − 2.87i)8-s − 0.992·10-s + (0.999 − 0.0100i)11-s + (0.377 + 0.274i)13-s + (0.221 − 0.681i)14-s + (−2.53 + 1.84i)16-s + (−1.09 + 0.797i)17-s + (0.308 − 0.950i)19-s + (1.09 + 0.797i)20-s + (−1.54 − 1.09i)22-s + 0.0249·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.828 + 0.560i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.828 + 0.560i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(693\)    =    \(3^{2} \cdot 7 \cdot 11\)
Sign: $0.828 + 0.560i$
Analytic conductor: \(5.53363\)
Root analytic conductor: \(2.35236\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{693} (631, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 693,\ (\ :1/2),\ 0.828 + 0.560i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.766090 - 0.234680i\)
\(L(\frac12)\) \(\approx\) \(0.766090 - 0.234680i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.309 - 0.951i)T \)
11 \( 1 + (-3.31 + 0.0332i)T \)
good2 \( 1 + (2.16 + 1.57i)T + (0.618 + 1.90i)T^{2} \)
5 \( 1 + (-0.946 + 0.687i)T + (1.54 - 4.75i)T^{2} \)
13 \( 1 + (-1.36 - 0.989i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (4.52 - 3.28i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-1.34 + 4.14i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 - 0.119T + 23T^{2} \)
29 \( 1 + (-1.35 - 4.18i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-5.10 - 3.71i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-1.80 - 5.56i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-3.40 + 10.4i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 5.97T + 43T^{2} \)
47 \( 1 + (3.88 - 11.9i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-0.337 - 0.245i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-0.223 - 0.688i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-7.16 + 5.20i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 - 8.78T + 67T^{2} \)
71 \( 1 + (-2.59 + 1.88i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (4.50 + 13.8i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (11.7 + 8.50i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (2.79 - 2.03i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 - 12.0T + 89T^{2} \)
97 \( 1 + (-4.85 - 3.52i)T + (29.9 + 92.2i)T^{2} \)
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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

−10.37281354811190948018766427285, −9.219979525033223109108552599301, −9.092539660443977554399264856631, −8.260477239793071898488347268690, −7.11805223836924605309123476807, −6.26568889539609276646858333811, −4.55334925614743502172767898110, −3.36627286710029670828603764490, −2.14938157239672827816254389845, −1.16091146925074487161667614600, 0.870390150104786361044980200208, 2.26417097098117036783469837799, 4.28669975876020865066540446818, 5.69609889909600503504544883253, 6.40103396870981102275331658760, 7.06926807278559182820884950947, 8.031971517775735823446822610128, 8.707417579188163167979818101866, 9.718669392792579032140081459985, 10.01556123185851806720302416348

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