Properties

Label 1-2205-2205.2167-r1-0-0
Degree $1$
Conductor $2205$
Sign $0.764 - 0.644i$
Analytic cond. $236.960$
Root an. cond. $236.960$
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  + (−0.433 − 0.900i)2-s + (−0.623 + 0.781i)4-s + (0.974 + 0.222i)8-s + (0.0747 − 0.997i)11-s + (−0.997 − 0.0747i)13-s + (−0.222 − 0.974i)16-s + (0.930 + 0.365i)17-s + (0.5 − 0.866i)19-s + (−0.930 + 0.365i)22-s + (−0.149 + 0.988i)23-s + (0.365 + 0.930i)26-s + (−0.365 + 0.930i)29-s + 31-s + (−0.781 + 0.623i)32-s + (−0.0747 − 0.997i)34-s + ⋯
L(s)  = 1  + (−0.433 − 0.900i)2-s + (−0.623 + 0.781i)4-s + (0.974 + 0.222i)8-s + (0.0747 − 0.997i)11-s + (−0.997 − 0.0747i)13-s + (−0.222 − 0.974i)16-s + (0.930 + 0.365i)17-s + (0.5 − 0.866i)19-s + (−0.930 + 0.365i)22-s + (−0.149 + 0.988i)23-s + (0.365 + 0.930i)26-s + (−0.365 + 0.930i)29-s + 31-s + (−0.781 + 0.623i)32-s + (−0.0747 − 0.997i)34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.764 - 0.644i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.764 - 0.644i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2205\)    =    \(3^{2} \cdot 5 \cdot 7^{2}\)
Sign: $0.764 - 0.644i$
Analytic conductor: \(236.960\)
Root analytic conductor: \(236.960\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2205} (2167, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2205,\ (1:\ ),\ 0.764 - 0.644i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.305860293 - 0.4768339659i\)
\(L(\frac12)\) \(\approx\) \(1.305860293 - 0.4768339659i\)
\(L(1)\) \(\approx\) \(0.7550681746 - 0.3065177166i\)
\(L(1)\) \(\approx\) \(0.7550681746 - 0.3065177166i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good2 \( 1 + (-0.433 - 0.900i)T \)
11 \( 1 + (0.0747 - 0.997i)T \)
13 \( 1 + (-0.997 - 0.0747i)T \)
17 \( 1 + (0.930 + 0.365i)T \)
19 \( 1 + (0.5 - 0.866i)T \)
23 \( 1 + (-0.149 + 0.988i)T \)
29 \( 1 + (-0.365 + 0.930i)T \)
31 \( 1 + T \)
37 \( 1 + (-0.149 - 0.988i)T \)
41 \( 1 + (-0.733 - 0.680i)T \)
43 \( 1 + (0.680 + 0.733i)T \)
47 \( 1 + (-0.433 - 0.900i)T \)
53 \( 1 + (-0.149 + 0.988i)T \)
59 \( 1 + (0.222 + 0.974i)T \)
61 \( 1 + (0.623 + 0.781i)T \)
67 \( 1 + iT \)
71 \( 1 + (0.623 - 0.781i)T \)
73 \( 1 + (0.997 - 0.0747i)T \)
79 \( 1 - T \)
83 \( 1 + (0.997 - 0.0747i)T \)
89 \( 1 + (-0.826 + 0.563i)T \)
97 \( 1 + (0.866 - 0.5i)T \)
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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

−19.40097608988312554528285532532, −18.760573653158648578659021805048, −18.18769358504189442229619578107, −17.141733077988913352618311300173, −17.02804357151478758238515555167, −16.006985335822983834489071104957, −15.38997003544369354975519610148, −14.48110963165188045528738940992, −14.28378363127238162859719231584, −13.19049360029752673973659241969, −12.343079748490409061645149678134, −11.70154082859427164722158995435, −10.43988543467863296030188210851, −9.7890155296408655051086980121, −9.50335224539531398061752936394, −8.1529496310826233621839520748, −7.85551513508244604052861659533, −6.91304329144591086410671020688, −6.320408250236229591193748341599, −5.23581074747282652420546925977, −4.76105521281747045016353147988, −3.7917036086871595393590080304, −2.50660870370524083325561509233, −1.52939122352769302206631019705, −0.46479210641329148860620735252, 0.60097373197798934422182256328, 1.39994372908466496953661725310, 2.5201890429508789381126940062, 3.21291024537207697646233092171, 3.96610307342960969820299368092, 5.05044090839194344786628517117, 5.68498675684792934815043890209, 7.03541345252911038956249063732, 7.66064697923125169003949787458, 8.50848221771786859402164405267, 9.22828024096457348508219006130, 9.92922789678501404007784049759, 10.65415276149754536983208649329, 11.42915284687714192255921673852, 12.03579552571106709690127478151, 12.74812328335541124201610586129, 13.61568581665299608419485339782, 14.13546459717719241583818152697, 15.09564013683091781605548921674, 16.11569773743389642538289540481, 16.74727556008057137455646166677, 17.45634334154145017490296739108, 18.07480137254502822941275187838, 18.97332907925689141396253721366, 19.46477909460010746062710860597

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