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Properties

Label 1-693-693.160-r0-0-0
Degree $1$
Conductor $693$
Sign $-0.710 + 0.703i$
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  + (−0.913 − 0.406i)2-s + (0.669 + 0.743i)4-s + (−0.913 + 0.406i)5-s + (−0.309 − 0.951i)8-s + 10-s + (−0.104 − 0.994i)13-s + (−0.104 + 0.994i)16-s + (−0.809 − 0.587i)17-s + (0.309 + 0.951i)19-s + (−0.913 − 0.406i)20-s + (−0.5 + 0.866i)23-s + (0.669 − 0.743i)25-s + (−0.309 + 0.951i)26-s + (0.978 − 0.207i)29-s + (0.104 + 0.994i)31-s + (0.5 − 0.866i)32-s + ⋯
L(s)  = 1  + (−0.913 − 0.406i)2-s + (0.669 + 0.743i)4-s + (−0.913 + 0.406i)5-s + (−0.309 − 0.951i)8-s + 10-s + (−0.104 − 0.994i)13-s + (−0.104 + 0.994i)16-s + (−0.809 − 0.587i)17-s + (0.309 + 0.951i)19-s + (−0.913 − 0.406i)20-s + (−0.5 + 0.866i)23-s + (0.669 − 0.743i)25-s + (−0.309 + 0.951i)26-s + (0.978 − 0.207i)29-s + (0.104 + 0.994i)31-s + (0.5 − 0.866i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.710 + 0.703i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.710 + 0.703i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(693\)    =    \(3^{2} \cdot 7 \cdot 11\)
Sign: $-0.710 + 0.703i$
Analytic conductor: \(3.21827\)
Root analytic conductor: \(3.21827\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{693} (160, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 693,\ (0:\ ),\ -0.710 + 0.703i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.07263654715 + 0.1764903446i\)
\(L(\frac12)\) \(\approx\) \(0.07263654715 + 0.1764903446i\)
\(L(1)\) \(\approx\) \(0.4961464675 + 0.01264106803i\)
\(L(1)\) \(\approx\) \(0.4961464675 + 0.01264106803i\)

Euler product

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

−22.47041131185394460059580454641, −21.454549725895908717098461763015, −20.3108887709620673704005499476, −19.89339577454795793567295156623, −19.036142330915556495722726361954, −18.396136416422338040557923355853, −17.31575019706803794160441407175, −16.68740308053356110338915808820, −15.837668188839149928269402579129, −15.28272380057725044370363215985, −14.35437338051014046288775105987, −13.26814729240331978378447196365, −12.02686071941840719834004529263, −11.45171551409079378228003838551, −10.567129105240827089875000576621, −9.50145753303096013016774856899, −8.6675537268534599894783760672, −8.08656777034248682090501348466, −6.98868186550718857204159166769, −6.39886788797099523859562215884, −4.9945830848400197697689379265, −4.19563031434456584609456439302, −2.71347573976907060904962621487, −1.5214200397571558926622328164, −0.1335631220189887012646808864, 1.310503628401838554617690094075, 2.725325252161988653206091678327, 3.4111917185048489409157310769, 4.51745191537637068376426544655, 6.00093690800126625818113631280, 7.08071370738723328290128409566, 7.79964178280618159737870130437, 8.47293721690279363755069417919, 9.57816702595631099491738197836, 10.43942165746833565665480691376, 11.15645361743108218098520766095, 11.99601801644689280994858801078, 12.64070186978733014245313322097, 13.86494576801166500994931255374, 15.01637604307330859839974658884, 15.83328104596840029497031839450, 16.28482312778959586987141836548, 17.649313348860407548519770351053, 17.994274152456054276037952276264, 18.99653310261149465078872140106, 19.71568995931997264587447459786, 20.23238122111550502269636135688, 21.16726840988187066914687390570, 22.16704719232926757372740300024, 22.84098419897869437084034581933

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