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Properties

Label 1-735-735.323-r0-0-0
Degree $1$
Conductor $735$
Sign $0.755 - 0.654i$
Analytic cond. $3.41332$
Root an. cond. $3.41332$
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  + (−0.781 + 0.623i)2-s + (0.222 − 0.974i)4-s + (0.433 + 0.900i)8-s + (−0.623 − 0.781i)11-s + (−0.781 + 0.623i)13-s + (−0.900 − 0.433i)16-s + (0.974 − 0.222i)17-s − 19-s + (0.974 + 0.222i)22-s + (0.974 + 0.222i)23-s + (0.222 − 0.974i)26-s + (−0.222 − 0.974i)29-s + 31-s + (0.974 − 0.222i)32-s + (−0.623 + 0.781i)34-s + ⋯
L(s)  = 1  + (−0.781 + 0.623i)2-s + (0.222 − 0.974i)4-s + (0.433 + 0.900i)8-s + (−0.623 − 0.781i)11-s + (−0.781 + 0.623i)13-s + (−0.900 − 0.433i)16-s + (0.974 − 0.222i)17-s − 19-s + (0.974 + 0.222i)22-s + (0.974 + 0.222i)23-s + (0.222 − 0.974i)26-s + (−0.222 − 0.974i)29-s + 31-s + (0.974 − 0.222i)32-s + (−0.623 + 0.781i)34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.755 - 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.755 - 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(735\)    =    \(3 \cdot 5 \cdot 7^{2}\)
Sign: $0.755 - 0.654i$
Analytic conductor: \(3.41332\)
Root analytic conductor: \(3.41332\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{735} (323, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 735,\ (0:\ ),\ 0.755 - 0.654i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6683996299 - 0.2492473980i\)
\(L(\frac12)\) \(\approx\) \(0.6683996299 - 0.2492473980i\)
\(L(1)\) \(\approx\) \(0.6756977338 + 0.05251581478i\)
\(L(1)\) \(\approx\) \(0.6756977338 + 0.05251581478i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good2 \( 1 + (-0.781 + 0.623i)T \)
11 \( 1 + (-0.623 - 0.781i)T \)
13 \( 1 + (-0.781 + 0.623i)T \)
17 \( 1 + (0.974 - 0.222i)T \)
19 \( 1 - T \)
23 \( 1 + (0.974 + 0.222i)T \)
29 \( 1 + (-0.222 - 0.974i)T \)
31 \( 1 + T \)
37 \( 1 + (-0.974 + 0.222i)T \)
41 \( 1 + (0.900 - 0.433i)T \)
43 \( 1 + (0.433 - 0.900i)T \)
47 \( 1 + (-0.781 + 0.623i)T \)
53 \( 1 + (0.974 + 0.222i)T \)
59 \( 1 + (-0.900 - 0.433i)T \)
61 \( 1 + (-0.222 - 0.974i)T \)
67 \( 1 - iT \)
71 \( 1 + (0.222 - 0.974i)T \)
73 \( 1 + (0.781 + 0.623i)T \)
79 \( 1 - T \)
83 \( 1 + (-0.781 - 0.623i)T \)
89 \( 1 + (0.623 - 0.781i)T \)
97 \( 1 - iT \)
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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

−22.628430589014772951592371569359, −21.34066542277782035290955887549, −21.08207102692881022808678173679, −20.034817459426085370685519107633, −19.39822162856861158871764383729, −18.60545810172778424690353160477, −17.74952409857351249001893814490, −17.13171628940128201866356039352, −16.3192686207141073499355867822, −15.2857062164059802877682070158, −14.54561189871861431461586918521, −13.09207872935025305938463709281, −12.62723947732541721318937885326, −11.83272550116575009802827844729, −10.6397092502284186186502218454, −10.22676036838651499162662638641, −9.29702532085229546994778083485, −8.31106802514854910620951533517, −7.55529892233797039350395116835, −6.74553055332711687963619490963, −5.319743322842508156545455495984, −4.32841353260763132824643670504, −3.093114046135666852717000827206, −2.33578895796227262334686208244, −1.10841932757562651936893555598, 0.49710430859518467417519585689, 1.8959235741427239434838827818, 2.98527821990693549354831361815, 4.51895254535704229102900544327, 5.43806610387419421086602491644, 6.30812839061761512488801768418, 7.2715521309348075253269752673, 8.041029939912380483603183568250, 8.89925156278021350265464789436, 9.75944401851742604428865708112, 10.57934605701058160535665166832, 11.395426569376964664535130173229, 12.4307548284054115323250854900, 13.64531902893998401858209449717, 14.31864909855825795913506254546, 15.25277395936061754916518578136, 15.92517473645780653437199462175, 17.00690357853519751732713956051, 17.17929559037700023329472412474, 18.52549341346032274057116240543, 19.03170797283875542774600344752, 19.595793176622325776172144241553, 20.92778252442155606621676026527, 21.32050623295277196851579311056, 22.71317311815203173543184984047

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