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Properties

Label 2-693-11.4-c1-0-26
Degree $2$
Conductor $693$
Sign $-0.999 + 0.0325i$
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  + (0.183 + 0.132i)2-s + (−0.602 − 1.85i)4-s + (−2.01 + 1.46i)5-s + (0.309 + 0.951i)7-s + (0.276 − 0.849i)8-s − 0.564·10-s + (2.66 − 1.97i)11-s + (−4.15 − 3.01i)13-s + (−0.0699 + 0.215i)14-s + (−2.98 + 2.17i)16-s + (−1.16 + 0.844i)17-s + (−1.87 + 5.77i)19-s + (3.93 + 2.85i)20-s + (0.750 − 0.00659i)22-s − 7.08·23-s + ⋯
L(s)  = 1  + (0.129 + 0.0940i)2-s + (−0.301 − 0.926i)4-s + (−0.902 + 0.655i)5-s + (0.116 + 0.359i)7-s + (0.0975 − 0.300i)8-s − 0.178·10-s + (0.803 − 0.594i)11-s + (−1.15 − 0.837i)13-s + (−0.0186 + 0.0574i)14-s + (−0.747 + 0.543i)16-s + (−0.282 + 0.204i)17-s + (−0.430 + 1.32i)19-s + (0.879 + 0.639i)20-s + (0.159 − 0.00140i)22-s − 1.47·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0325i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0325i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(693\)    =    \(3^{2} \cdot 7 \cdot 11\)
Sign: $-0.999 + 0.0325i$
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: \((2,\ 693,\ (\ :1/2),\ -0.999 + 0.0325i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00209558 - 0.128714i\)
\(L(\frac12)\) \(\approx\) \(0.00209558 - 0.128714i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.66 + 1.97i)T \)
good2 \( 1 + (-0.183 - 0.132i)T + (0.618 + 1.90i)T^{2} \)
5 \( 1 + (2.01 - 1.46i)T + (1.54 - 4.75i)T^{2} \)
13 \( 1 + (4.15 + 3.01i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (1.16 - 0.844i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (1.87 - 5.77i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + 7.08T + 23T^{2} \)
29 \( 1 + (2.01 + 6.19i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (6.22 + 4.51i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (1.23 + 3.78i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (2.08 - 6.41i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 0.802T + 43T^{2} \)
47 \( 1 + (2.08 - 6.42i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-5.32 - 3.86i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (0.888 + 2.73i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-0.691 + 0.502i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + 1.64T + 67T^{2} \)
71 \( 1 + (-3.65 + 2.65i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (4.58 + 14.1i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (1.98 + 1.44i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-1.81 + 1.32i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + 1.73T + 89T^{2} \)
97 \( 1 + (-9.77 - 7.09i)T + (29.9 + 92.2i)T^{2} \)
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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

−10.06507491731536067071844657888, −9.331807243922318950289728518739, −8.156257284375563873421612684532, −7.50381377868132041579950766258, −6.23685991520938473064640110860, −5.71693051147059081502265432133, −4.39115417714864633834381833809, −3.56607871501856589889877774450, −2.00736597968515148287647716497, −0.06211053370300269099319149531, 2.11122368624654462820966833509, 3.66950059820888842696428177278, 4.37498648445733463735319080051, 5.02174525471618439992811165167, 7.00879435627334062149179762775, 7.20344349326745189130400927345, 8.448372640695146820183419015932, 8.941143292291558286756588848566, 9.884328891776014077136354070901, 11.18004333278289520887769453830

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