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Properties

Label 2-693-11.4-c1-0-21
Degree $2$
Conductor $693$
Sign $0.970 - 0.242i$
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  + (1.43 + 1.04i)2-s + (0.352 + 1.08i)4-s + (0.477 − 0.346i)5-s + (−0.309 − 0.951i)7-s + (0.470 − 1.44i)8-s + 1.04·10-s + (2.19 − 2.48i)11-s + (1.24 + 0.907i)13-s + (0.547 − 1.68i)14-s + (4.02 − 2.92i)16-s + (5.78 − 4.19i)17-s + (−1.91 + 5.88i)19-s + (0.544 + 0.395i)20-s + (5.73 − 1.28i)22-s − 3.76·23-s + ⋯
L(s)  = 1  + (1.01 + 0.736i)2-s + (0.176 + 0.542i)4-s + (0.213 − 0.155i)5-s + (−0.116 − 0.359i)7-s + (0.166 − 0.512i)8-s + 0.330·10-s + (0.660 − 0.750i)11-s + (0.346 + 0.251i)13-s + (0.146 − 0.450i)14-s + (1.00 − 0.731i)16-s + (1.40 − 1.01i)17-s + (−0.438 + 1.35i)19-s + (0.121 + 0.0884i)20-s + (1.22 − 0.274i)22-s − 0.784·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 - 0.242i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.970 - 0.242i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(693\)    =    \(3^{2} \cdot 7 \cdot 11\)
Sign: $0.970 - 0.242i$
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: \((2,\ 693,\ (\ :1/2),\ 0.970 - 0.242i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.71445 + 0.333678i\)
\(L(\frac12)\) \(\approx\) \(2.71445 + 0.333678i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.19 + 2.48i)T \)
good2 \( 1 + (-1.43 - 1.04i)T + (0.618 + 1.90i)T^{2} \)
5 \( 1 + (-0.477 + 0.346i)T + (1.54 - 4.75i)T^{2} \)
13 \( 1 + (-1.24 - 0.907i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-5.78 + 4.19i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (1.91 - 5.88i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + 3.76T + 23T^{2} \)
29 \( 1 + (0.187 + 0.577i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-5.55 - 4.03i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (2.38 + 7.32i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (2.31 - 7.11i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 10.9T + 43T^{2} \)
47 \( 1 + (-3.15 + 9.70i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-5.69 - 4.14i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-1.03 - 3.18i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-2.72 + 1.98i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + 2.19T + 67T^{2} \)
71 \( 1 + (11.2 - 8.19i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (0.800 + 2.46i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-3.77 - 2.74i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (2.83 - 2.06i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + 8.15T + 89T^{2} \)
97 \( 1 + (-3.88 - 2.81i)T + (29.9 + 92.2i)T^{2} \)
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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

−10.30713530710174031523590654495, −9.784382642252231752246772918241, −8.605682526198494298172515138278, −7.63792626648313532899347784018, −6.73438241121378594999184068122, −5.89913787964078471397549337750, −5.25920132426080555494260257875, −4.01772440133378043496581810776, −3.35549655929745692459447727869, −1.27512513174240343490059066463, 1.71166455962195495957697892666, 2.82294609292739779744409063645, 3.85838998442792278139075027291, 4.72157605209826263957872080846, 5.77969096542917425635173648710, 6.58271612217081621645655334942, 7.914321515309664878368300569351, 8.692818233038916738535493807243, 9.951631513458584927290162386151, 10.45197168163765813281340267010

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