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Properties

Label 2-693-11.9-c1-0-18
Degree $2$
Conductor $693$
Sign $0.959 + 0.283i$
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  + (−0.206 + 0.636i)2-s + (1.25 + 0.911i)4-s + (−0.662 − 2.03i)5-s + (−0.809 − 0.587i)7-s + (−1.92 + 1.39i)8-s + 1.43·10-s + (1.08 − 3.13i)11-s + (0.781 − 2.40i)13-s + (0.541 − 0.393i)14-s + (0.466 + 1.43i)16-s + (0.553 + 1.70i)17-s + (5.44 − 3.95i)19-s + (1.02 − 3.16i)20-s + (1.77 + 1.33i)22-s + 3.16·23-s + ⋯
L(s)  = 1  + (−0.146 + 0.450i)2-s + (0.627 + 0.455i)4-s + (−0.296 − 0.911i)5-s + (−0.305 − 0.222i)7-s + (−0.680 + 0.494i)8-s + 0.454·10-s + (0.326 − 0.945i)11-s + (0.216 − 0.666i)13-s + (0.144 − 0.105i)14-s + (0.116 + 0.358i)16-s + (0.134 + 0.413i)17-s + (1.24 − 0.907i)19-s + (0.229 − 0.707i)20-s + (0.377 + 0.285i)22-s + 0.659·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.959 + 0.283i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.959 + 0.283i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(693\)    =    \(3^{2} \cdot 7 \cdot 11\)
Sign: $0.959 + 0.283i$
Analytic conductor: \(5.53363\)
Root analytic conductor: \(2.35236\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{693} (64, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 693,\ (\ :1/2),\ 0.959 + 0.283i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.48150 - 0.214321i\)
\(L(\frac12)\) \(\approx\) \(1.48150 - 0.214321i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.809 + 0.587i)T \)
11 \( 1 + (-1.08 + 3.13i)T \)
good2 \( 1 + (0.206 - 0.636i)T + (-1.61 - 1.17i)T^{2} \)
5 \( 1 + (0.662 + 2.03i)T + (-4.04 + 2.93i)T^{2} \)
13 \( 1 + (-0.781 + 2.40i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (-0.553 - 1.70i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (-5.44 + 3.95i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 - 3.16T + 23T^{2} \)
29 \( 1 + (0.747 + 0.543i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-0.927 + 2.85i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-1.21 - 0.885i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-4.49 + 3.26i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 8.42T + 43T^{2} \)
47 \( 1 + (-3.55 + 2.58i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (0.206 - 0.634i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (0.298 + 0.216i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-1.54 - 4.76i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + 0.902T + 67T^{2} \)
71 \( 1 + (-4.59 - 14.1i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (6.50 + 4.72i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (-1.25 + 3.85i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-1.25 - 3.85i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 - 8.30T + 89T^{2} \)
97 \( 1 + (-2.63 + 8.09i)T + (-78.4 - 57.0i)T^{2} \)
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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

−10.50048913792684968291548868927, −9.253960445390022731049298752520, −8.593147757518732230236137269481, −7.85820493988296098101543108083, −6.99740108810559114990516714093, −6.01432016041331703129380277862, −5.15049621962219509439977236481, −3.76765841492920552523292351667, −2.84046279342465533992178435930, −0.907668805281116667998653514345, 1.49334885507516025453478534445, 2.75860037067622386924619303465, 3.61246668731982365316725328553, 5.09173609989272604295924433846, 6.27518795510468493775249243909, 6.94651992657314442911794836986, 7.63919029376379565666660648585, 9.156178202799021190879288029265, 9.758836014618382188680164502371, 10.52196217186931357154617036836

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