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Properties

Label 1-693-693.139-r0-0-0
Degree $1$
Conductor $693$
Sign $0.994 + 0.104i$
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  + (0.978 + 0.207i)2-s + (0.913 + 0.406i)4-s + (0.978 − 0.207i)5-s + (0.809 + 0.587i)8-s + 10-s + (0.669 − 0.743i)13-s + (0.669 + 0.743i)16-s + (0.309 − 0.951i)17-s + (−0.809 − 0.587i)19-s + (0.978 + 0.207i)20-s + (−0.5 − 0.866i)23-s + (0.913 − 0.406i)25-s + (0.809 − 0.587i)26-s + (0.104 + 0.994i)29-s + (−0.669 + 0.743i)31-s + (0.5 + 0.866i)32-s + ⋯
L(s)  = 1  + (0.978 + 0.207i)2-s + (0.913 + 0.406i)4-s + (0.978 − 0.207i)5-s + (0.809 + 0.587i)8-s + 10-s + (0.669 − 0.743i)13-s + (0.669 + 0.743i)16-s + (0.309 − 0.951i)17-s + (−0.809 − 0.587i)19-s + (0.978 + 0.207i)20-s + (−0.5 − 0.866i)23-s + (0.913 − 0.406i)25-s + (0.809 − 0.587i)26-s + (0.104 + 0.994i)29-s + (−0.669 + 0.743i)31-s + (0.5 + 0.866i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.994 + 0.104i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.994 + 0.104i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(693\)    =    \(3^{2} \cdot 7 \cdot 11\)
Sign: $0.994 + 0.104i$
Analytic conductor: \(3.21827\)
Root analytic conductor: \(3.21827\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{693} (139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 693,\ (0:\ ),\ 0.994 + 0.104i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.287685947 + 0.1727612488i\)
\(L(\frac12)\) \(\approx\) \(3.287685947 + 0.1727612488i\)
\(L(1)\) \(\approx\) \(2.239908734 + 0.1498800839i\)
\(L(1)\) \(\approx\) \(2.239908734 + 0.1498800839i\)

Euler product

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

−22.61920750891218258564063429400, −21.70047030687649263879490739140, −21.2136746050238482616072355556, −20.620654590183846511167446886712, −19.39100159779084354901046722623, −18.845067293683744795553754145896, −17.66803540190110700296226725425, −16.829817479365749513507968553143, −16.00056572726707846687837726957, −14.949795461943155130209900790235, −14.31668483782519872116003670044, −13.51235673966195358097256339602, −12.89724155704829269575139036602, −11.91454329581411890198408999817, −10.9789727454255128584350248040, −10.24667250094860138544002908246, −9.37576869988910668437888195746, −8.12150512639492419447338250749, −6.90919325758338002686504466672, −6.06222396088647521518425905760, −5.56490947626119868376857475701, −4.23844682624331233575880471031, −3.5015514728284213711627251161, −2.14315995232039463604823977777, −1.59362849958098197500200156460, 1.35365301087204146301922415931, 2.49586391665703685077596615639, 3.34018292329332334622079727905, 4.653409363711673107499457547779, 5.33757814954631043660791864533, 6.246022399861904464898497471716, 6.96705475978490963582442436645, 8.18169062350160141322527747987, 9.08089093779208639666617378059, 10.35815331361592414388180495934, 10.922339716859829627157317483241, 12.17332005890289892471083778253, 12.836524328576352071004935914167, 13.623241167487482717416854902097, 14.26026242490766357674297985034, 15.145651468506053160247113495648, 16.10498974686826102229692907854, 16.73807330422269540532663887086, 17.71552760090339120472703019895, 18.42743746255005525965216705649, 19.80021655425968087842039881997, 20.525267695231047100249146340991, 21.12837012188667356935544281267, 21.96939696227396878891518879857, 22.58254821811555984885104301275

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