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Properties

Label 1-1205-1205.1014-r0-0-0
Degree $1$
Conductor $1205$
Sign $0.350 - 0.936i$
Analytic cond. $5.59599$
Root an. cond. $5.59599$
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  + (−0.866 + 0.5i)2-s + (0.743 − 0.669i)3-s + (0.5 − 0.866i)4-s + (−0.309 + 0.951i)6-s + (−0.933 + 0.358i)7-s + i·8-s + (0.104 − 0.994i)9-s + (−0.965 − 0.258i)11-s + (−0.207 − 0.978i)12-s + (0.0523 − 0.998i)13-s + (0.629 − 0.777i)14-s + (−0.5 − 0.866i)16-s + (0.987 + 0.156i)17-s + (0.406 + 0.913i)18-s + (0.965 + 0.258i)19-s + ⋯
L(s)  = 1  + (−0.866 + 0.5i)2-s + (0.743 − 0.669i)3-s + (0.5 − 0.866i)4-s + (−0.309 + 0.951i)6-s + (−0.933 + 0.358i)7-s + i·8-s + (0.104 − 0.994i)9-s + (−0.965 − 0.258i)11-s + (−0.207 − 0.978i)12-s + (0.0523 − 0.998i)13-s + (0.629 − 0.777i)14-s + (−0.5 − 0.866i)16-s + (0.987 + 0.156i)17-s + (0.406 + 0.913i)18-s + (0.965 + 0.258i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.350 - 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.350 - 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1205\)    =    \(5 \cdot 241\)
Sign: $0.350 - 0.936i$
Analytic conductor: \(5.59599\)
Root analytic conductor: \(5.59599\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1205} (1014, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1205,\ (0:\ ),\ 0.350 - 0.936i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8554004009 - 0.5930875699i\)
\(L(\frac12)\) \(\approx\) \(0.8554004009 - 0.5930875699i\)
\(L(1)\) \(\approx\) \(0.8235899207 - 0.1371291960i\)
\(L(1)\) \(\approx\) \(0.8235899207 - 0.1371291960i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
241 \( 1 \)
good2 \( 1 + (-0.866 + 0.5i)T \)
3 \( 1 + (0.743 - 0.669i)T \)
7 \( 1 + (-0.933 + 0.358i)T \)
11 \( 1 + (-0.965 - 0.258i)T \)
13 \( 1 + (0.0523 - 0.998i)T \)
17 \( 1 + (0.987 + 0.156i)T \)
19 \( 1 + (0.965 + 0.258i)T \)
23 \( 1 + (0.453 + 0.891i)T \)
29 \( 1 + (0.994 + 0.104i)T \)
31 \( 1 + (0.358 + 0.933i)T \)
37 \( 1 + (0.0523 + 0.998i)T \)
41 \( 1 + (-0.587 - 0.809i)T \)
43 \( 1 + (-0.156 - 0.987i)T \)
47 \( 1 + (0.587 - 0.809i)T \)
53 \( 1 + (-0.406 - 0.913i)T \)
59 \( 1 + (0.743 + 0.669i)T \)
61 \( 1 + (-0.951 - 0.309i)T \)
67 \( 1 + (0.406 - 0.913i)T \)
71 \( 1 + (-0.777 + 0.629i)T \)
73 \( 1 + (-0.891 - 0.453i)T \)
79 \( 1 + (-0.951 + 0.309i)T \)
83 \( 1 + (0.669 - 0.743i)T \)
89 \( 1 + (0.258 - 0.965i)T \)
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

−21.07287083336409806073784609257, −20.49818404819692060179602629042, −19.83660204550952509279615180550, −18.96375038447016833344880812253, −18.66126684739449769799936648419, −17.496884894966375185157193845513, −16.39945420120330118932948752392, −16.25803021368364297373744511171, −15.42220342092288414721752897756, −14.327396315023024742740995561836, −13.46974434602008879979577499443, −12.76266499611502092666418340651, −11.79883241878008284538606122188, −10.810670101811886820880831989364, −10.09065886606205138001733646307, −9.58551601116193188152852548500, −8.89099672051076919858592985958, −7.86557493591199672029974862612, −7.32330584702981410423765179942, −6.24604432430013704612044604002, −4.8192386633440602083574860329, −3.937608521941222864984285211, −2.95560190482514425715066809066, −2.51722117019864970351722417845, −1.094063862240305039031969853044, 0.58161780340692306394559442400, 1.62838942648601672498142299176, 2.95641904595040626251705882148, 3.219913497244207270549016925417, 5.2720261511633968560972417743, 5.83226034784361598219465688357, 6.87639879220003282939937311186, 7.53708471819436348130188852476, 8.29572642926047904732906963052, 8.918887626402215630823351275282, 10.033132863467131082403637088206, 10.25250927910044060216130064797, 11.74567592303073497961314941688, 12.45705328547222798194249834649, 13.42543122257037219394329463319, 14.01359516393076277571381939439, 15.151323817056919129018031811191, 15.58260881563312420046488603614, 16.30969487201117163815984254206, 17.38842828440811551242358295388, 18.09411384299871640025173418256, 18.78522727228615522509560324493, 19.22722495272380913715287197737, 20.087501831382305885531750132754, 20.64627942735212842221255542889

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