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Properties

Label 2-693-11.4-c1-0-19
Degree $2$
Conductor $693$
Sign $-0.703 + 0.710i$
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  + (−1.38 − 1.00i)2-s + (0.282 + 0.869i)4-s + (3.28 − 2.39i)5-s + (0.309 + 0.951i)7-s + (−0.572 + 1.76i)8-s − 6.94·10-s + (0.582 − 3.26i)11-s + (−2.65 − 1.92i)13-s + (0.527 − 1.62i)14-s + (4.03 − 2.93i)16-s + (−1.06 + 0.776i)17-s + (0.668 − 2.05i)19-s + (3.00 + 2.18i)20-s + (−4.08 + 3.92i)22-s + 1.86·23-s + ⋯
L(s)  = 1  + (−0.976 − 0.709i)2-s + (0.141 + 0.434i)4-s + (1.47 − 1.06i)5-s + (0.116 + 0.359i)7-s + (−0.202 + 0.623i)8-s − 2.19·10-s + (0.175 − 0.984i)11-s + (−0.735 − 0.534i)13-s + (0.140 − 0.433i)14-s + (1.00 − 0.733i)16-s + (−0.259 + 0.188i)17-s + (0.153 − 0.471i)19-s + (0.672 + 0.488i)20-s + (−0.869 + 0.836i)22-s + 0.389·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.703 + 0.710i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.703 + 0.710i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(693\)    =    \(3^{2} \cdot 7 \cdot 11\)
Sign: $-0.703 + 0.710i$
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: \((2,\ 693,\ (\ :1/2),\ -0.703 + 0.710i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.402839 - 0.966458i\)
\(L(\frac12)\) \(\approx\) \(0.402839 - 0.966458i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.582 + 3.26i)T \)
good2 \( 1 + (1.38 + 1.00i)T + (0.618 + 1.90i)T^{2} \)
5 \( 1 + (-3.28 + 2.39i)T + (1.54 - 4.75i)T^{2} \)
13 \( 1 + (2.65 + 1.92i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (1.06 - 0.776i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-0.668 + 2.05i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 - 1.86T + 23T^{2} \)
29 \( 1 + (-0.0754 - 0.232i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-5.55 - 4.03i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-0.0789 - 0.243i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-1.77 + 5.45i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 8.01T + 43T^{2} \)
47 \( 1 + (-1.25 + 3.86i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (4.04 + 2.94i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-0.303 - 0.935i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (1.49 - 1.08i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + 3.00T + 67T^{2} \)
71 \( 1 + (-5.23 + 3.80i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (2.98 + 9.17i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-4.47 - 3.25i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-1.67 + 1.21i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + 16.6T + 89T^{2} \)
97 \( 1 + (-2.09 - 1.51i)T + (29.9 + 92.2i)T^{2} \)
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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

−10.06867594063312543702260310589, −9.242717153569871756933941078510, −8.798384156853712907975672809817, −8.061712721707388887297271317281, −6.46292451550577606778306400535, −5.48790736372671827160497503232, −4.93787247628987397372210569994, −2.91948759897888295279114525296, −1.91159391296781962544641443635, −0.78667343189598338377842745382, 1.68462006511209760999439951442, 2.90788756375022975682770832626, 4.47765023796395288810685053359, 5.82279512293050621838654881756, 6.73764072543600026803546243327, 7.09965038634368850296554763183, 8.091486684686911218779936771397, 9.357754749231296473605565448563, 9.749805223205660040492553085256, 10.28340046913383114980648292791

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