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Properties

Label 2-693-11.3-c1-0-21
Degree $2$
Conductor $693$
Sign $0.642 + 0.766i$
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  + (−0.5 + 0.363i)2-s + (−0.5 + 1.53i)4-s + (−0.309 − 0.224i)5-s + (0.309 − 0.951i)7-s + (−0.690 − 2.12i)8-s + 0.236·10-s + (−0.309 − 3.30i)11-s + (0.809 − 0.587i)13-s + (0.190 + 0.587i)14-s + (−1.49 − 1.08i)16-s + (−3.42 − 2.48i)17-s + (0.5 − 0.363i)20-s + (1.35 + 1.53i)22-s + 3.23·23-s + (−1.5 − 4.61i)25-s + (−0.190 + 0.587i)26-s + ⋯
L(s)  = 1  + (−0.353 + 0.256i)2-s + (−0.250 + 0.769i)4-s + (−0.138 − 0.100i)5-s + (0.116 − 0.359i)7-s + (−0.244 − 0.751i)8-s + 0.0746·10-s + (−0.0931 − 0.995i)11-s + (0.224 − 0.163i)13-s + (0.0510 + 0.157i)14-s + (−0.374 − 0.272i)16-s + (−0.831 − 0.603i)17-s + (0.111 − 0.0812i)20-s + (0.288 + 0.328i)22-s + 0.674·23-s + (−0.300 − 0.923i)25-s + (−0.0374 + 0.115i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.642 + 0.766i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.642 + 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(693\)    =    \(3^{2} \cdot 7 \cdot 11\)
Sign: $0.642 + 0.766i$
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: \((2,\ 693,\ (\ :1/2),\ 0.642 + 0.766i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.800948 - 0.373624i\)
\(L(\frac12)\) \(\approx\) \(0.800948 - 0.373624i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.309 + 3.30i)T \)
good2 \( 1 + (0.5 - 0.363i)T + (0.618 - 1.90i)T^{2} \)
5 \( 1 + (0.309 + 0.224i)T + (1.54 + 4.75i)T^{2} \)
13 \( 1 + (-0.809 + 0.587i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (3.42 + 2.48i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-15.3 + 11.1i)T^{2} \)
23 \( 1 - 3.23T + 23T^{2} \)
29 \( 1 + (2.07 - 6.37i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-8.28 + 6.01i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-2.14 + 6.60i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (1.57 + 4.84i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + T + 43T^{2} \)
47 \( 1 + (-2.26 - 6.96i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-6.16 + 4.47i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-1.28 + 3.94i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-4.66 - 3.38i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 9.23T + 67T^{2} \)
71 \( 1 + (6.04 + 4.39i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-3.57 + 10.9i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (8.78 - 6.37i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (4.85 + 3.52i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 - 6.38T + 89T^{2} \)
97 \( 1 + (-13.7 + 9.99i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.31790855487278424014791507338, −9.188401936647555327318239001993, −8.625243314756290111304122599665, −7.81243431599229907677910197323, −7.01032964050144868795948051171, −6.03304673580676565245667378003, −4.71411613428950864728825921064, −3.78193748237233534474475280577, −2.70461031854597476297427667824, −0.55808287327551655038774660347, 1.45107482186004068518253386746, 2.60042228517295993933452035322, 4.24801934825283045988662266753, 5.08852138783178100142832237390, 6.13515488110428919791766209177, 7.02004833729126138474054788288, 8.239682595456951553985256133482, 8.941358872247539193325722400329, 9.827449566951611664492266378664, 10.41353926471982110834346376509

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