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Properties

Label 1-693-693.394-r0-0-0
Degree $1$
Conductor $693$
Sign $0.346 - 0.938i$
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 + (−0.809 + 0.587i)5-s + (0.309 − 0.951i)8-s + (−0.5 + 0.866i)10-s + (0.913 − 0.406i)13-s + (−0.104 − 0.994i)16-s + (−0.104 − 0.994i)17-s + (−0.978 + 0.207i)19-s + (−0.104 + 0.994i)20-s + 23-s + (0.309 − 0.951i)25-s + (0.669 − 0.743i)26-s + (0.669 − 0.743i)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 + (−0.809 + 0.587i)5-s + (0.309 − 0.951i)8-s + (−0.5 + 0.866i)10-s + (0.913 − 0.406i)13-s + (−0.104 − 0.994i)16-s + (−0.104 − 0.994i)17-s + (−0.978 + 0.207i)19-s + (−0.104 + 0.994i)20-s + 23-s + (0.309 − 0.951i)25-s + (0.669 − 0.743i)26-s + (0.669 − 0.743i)29-s + (−0.104 + 0.994i)31-s + (−0.5 − 0.866i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.346 - 0.938i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.346 - 0.938i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(693\)    =    \(3^{2} \cdot 7 \cdot 11\)
Sign: $0.346 - 0.938i$
Analytic conductor: \(3.21827\)
Root analytic conductor: \(3.21827\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{693} (394, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 693,\ (0:\ ),\ 0.346 - 0.938i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.798070396 - 1.253116154i\)
\(L(\frac12)\) \(\approx\) \(1.798070396 - 1.253116154i\)
\(L(1)\) \(\approx\) \(1.543092972 - 0.5279193140i\)
\(L(1)\) \(\approx\) \(1.543092972 - 0.5279193140i\)

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 \( 1 + (-0.809 + 0.587i)T \)
13 \( 1 + (0.913 - 0.406i)T \)
17 \( 1 + (-0.104 - 0.994i)T \)
19 \( 1 + (-0.978 + 0.207i)T \)
23 \( 1 + T \)
29 \( 1 + (0.669 - 0.743i)T \)
31 \( 1 + (-0.104 + 0.994i)T \)
37 \( 1 + (0.669 - 0.743i)T \)
41 \( 1 + (0.669 + 0.743i)T \)
43 \( 1 + (-0.5 - 0.866i)T \)
47 \( 1 + (0.669 + 0.743i)T \)
53 \( 1 + (0.913 - 0.406i)T \)
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 \( 1 + (-0.978 - 0.207i)T \)
79 \( 1 + (0.913 - 0.406i)T \)
83 \( 1 + (0.913 + 0.406i)T \)
89 \( 1 + (-0.5 - 0.866i)T \)
97 \( 1 + (-0.104 + 0.994i)T \)
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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

−23.12559089301092915541532927272, −22.064776636365187090232168771774, −21.225287507021481840261154578278, −20.63250207530066797332277269, −19.698459344569216554893614208684, −18.985113833103469036898740327536, −17.69151144214389718394695106884, −16.70489388427980322310774790201, −16.2994277976597373103474191845, −15.1540630524389482543493664080, −14.915106847941956127593198356711, −13.52278693745110886585846528594, −12.99879044353149365878895419316, −12.15992230994328202613706970107, −11.32068502934993740735485415229, −10.598891154491219078943894140273, −8.86509387199948450001872631387, −8.39304534066193390704413085534, −7.34993605492453884135116661166, −6.45287873952278115566327094077, −5.533821017190706615705549358991, −4.385805605372028820683681878582, −3.94627464987155958879101847368, −2.76373388936548926372758673642, −1.37103172288662011057920857659, 0.864080134093522788289704962515, 2.39062157668879141888766140463, 3.23418105629946379628748992163, 4.0858800387077225230930390819, 4.9929575850710368421710443228, 6.16380233119397515743286538417, 6.89760624085658567676492003580, 7.86678080486136277437633249317, 9.02357609419326959342145755514, 10.3189449403534269721011460060, 10.95135455024735313136126781367, 11.64133894563765129937530498380, 12.528229849358372255011644073135, 13.36262440899573179664737856753, 14.25526538199397217474653856109, 15.015266430640771421424404827424, 15.73407595989786376509816319216, 16.393947101317211577751244396254, 17.81370072070080822535193829318, 18.736037753585776819532106617017, 19.34550472234071180192083927530, 20.1851761784255327012491615181, 20.95558513150632196001341658815, 21.7272632713930911943914336104, 22.73674878011173790109328753169

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