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Properties

Label 2-693-11.3-c1-0-28
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: $\chi_{693} (190, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 693,\ (\ :1/2),\ -0.828 + 0.560i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.918480 - 2.99828i\)
\(L(\frac12)\) \(\approx\) \(0.918480 - 2.99828i\)
\(L(\frac{3}{2})\) 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

−10.43456629100307035545229956317, −9.828067183980883860507831049657, −8.333612452204028055755268515375, −7.38217804768617994533733249770, −5.92834819099885328646929105730, −5.45845694671549416157900952816, −4.27817821690862066355311471453, −3.64001162326336721704421699662, −2.50203757762851506923560653470, −1.09377973920724194085332296775, 2.69054672421491056727607819540, 3.43988526657839576073359426644, 4.69202980812486479924649405200, 5.31450409395628465282702303998, 6.24964997455371435532522560200, 7.20356123080026971075232754109, 7.79652840762197353142686221722, 8.624857981883990835169712328150, 9.971328944750166260209933338908, 11.40714988926095565013605343901

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