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Properties

Label 1-693-693.556-r0-0-0
Degree $1$
Conductor $693$
Sign $-0.923 - 0.383i$
Analytic cond. $3.21827$
Root an. cond. $3.21827$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.809 − 0.587i)2-s + (0.309 − 0.951i)4-s + (0.104 − 0.994i)5-s + (−0.309 − 0.951i)8-s + (−0.5 − 0.866i)10-s + (−0.104 − 0.994i)13-s + (−0.809 − 0.587i)16-s + (−0.104 + 0.994i)17-s + (−0.978 − 0.207i)19-s + (−0.913 − 0.406i)20-s + (−0.5 − 0.866i)23-s + (−0.978 − 0.207i)25-s + (−0.669 − 0.743i)26-s + (0.978 − 0.207i)29-s + (0.809 − 0.587i)31-s − 32-s + ⋯
L(s)  = 1  + (0.809 − 0.587i)2-s + (0.309 − 0.951i)4-s + (0.104 − 0.994i)5-s + (−0.309 − 0.951i)8-s + (−0.5 − 0.866i)10-s + (−0.104 − 0.994i)13-s + (−0.809 − 0.587i)16-s + (−0.104 + 0.994i)17-s + (−0.978 − 0.207i)19-s + (−0.913 − 0.406i)20-s + (−0.5 − 0.866i)23-s + (−0.978 − 0.207i)25-s + (−0.669 − 0.743i)26-s + (0.978 − 0.207i)29-s + (0.809 − 0.587i)31-s − 32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.923 - 0.383i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.923 - 0.383i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(693\)    =    \(3^{2} \cdot 7 \cdot 11\)
Sign: $-0.923 - 0.383i$
Analytic conductor: \(3.21827\)
Root analytic conductor: \(3.21827\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{693} (556, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 693,\ (0:\ ),\ -0.923 - 0.383i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3782449682 - 1.899663983i\)
\(L(\frac12)\) \(\approx\) \(0.3782449682 - 1.899663983i\)
\(L(1)\) \(\approx\) \(1.131126199 - 1.017723166i\)
\(L(1)\) \(\approx\) \(1.131126199 - 1.017723166i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.809 - 0.587i)T \)
5 \( 1 + (0.104 - 0.994i)T \)
13 \( 1 + (-0.104 - 0.994i)T \)
17 \( 1 + (-0.104 + 0.994i)T \)
19 \( 1 + (-0.978 - 0.207i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
29 \( 1 + (0.978 - 0.207i)T \)
31 \( 1 + (0.809 - 0.587i)T \)
37 \( 1 + (0.669 + 0.743i)T \)
41 \( 1 + (-0.978 - 0.207i)T \)
43 \( 1 + (0.5 + 0.866i)T \)
47 \( 1 + (-0.309 - 0.951i)T \)
53 \( 1 + (0.913 + 0.406i)T \)
59 \( 1 + (-0.309 + 0.951i)T \)
61 \( 1 + (-0.809 - 0.587i)T \)
67 \( 1 + T \)
71 \( 1 + (-0.809 - 0.587i)T \)
73 \( 1 + (-0.978 + 0.207i)T \)
79 \( 1 + (0.809 - 0.587i)T \)
83 \( 1 + (-0.104 + 0.994i)T \)
89 \( 1 + (0.5 - 0.866i)T \)
97 \( 1 + (-0.913 - 0.406i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−23.2219756063973257605258134708, −22.211401761219584641492638204517, −21.60935480732039297257550518381, −20.975725674155027657554190159424, −19.79733074242129815603871908419, −18.92196152573994828624212015422, −17.95892384898643404843374116867, −17.27133386088893375073758985043, −16.22646828677995477301934592689, −15.57950689764277168025087825799, −14.63554053185260059401324277408, −14.043624020721229597644316353946, −13.40323219308741118656557428486, −12.16719955568988392851385957916, −11.55534472107917511685650741621, −10.62905461123640854069441524821, −9.50543743418862657000524463661, −8.410039025174083451431905929420, −7.363616069274552922437632764719, −6.71064634290137877514169549972, −5.9701365133267135543849832913, −4.8102649906587377899484761221, −3.91182999857338557649146016358, −2.900797069655295455552263641967, −2.01771626149227226799660418801, 0.678997568237548624165561765393, 1.8545854262170209475299411112, 2.87761336833993070097640054806, 4.15151894950974310077572372266, 4.723485235782454199703649223649, 5.805178185568899862290688991067, 6.46130370280423675527285071651, 8.00877116612437521719699789163, 8.7568715707694492995936142974, 10.00284728679806007096035840409, 10.50443477376812891249135551592, 11.69506287099510622012096931425, 12.46652646555696051493771702604, 13.04924149313972120579640376137, 13.77281272155158697423117994279, 14.94426833960160747098922979782, 15.45390811016049624329304950926, 16.53185953438016979078661923654, 17.344729992427212668456905845275, 18.36888427344910260085633677222, 19.477574246342757860402415958568, 19.96081217250818597343034469644, 20.7798588646362963634045789973, 21.44771886041533692217445971467, 22.20307875030323841110267550457

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