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Properties

Label 1-2205-2205.277-r1-0-0
Degree $1$
Conductor $2205$
Sign $-0.684 - 0.729i$
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  + (−0.781 + 0.623i)2-s + (0.222 − 0.974i)4-s + (0.433 + 0.900i)8-s + (−0.988 + 0.149i)11-s + (0.149 + 0.988i)13-s + (−0.900 − 0.433i)16-s + (−0.680 − 0.733i)17-s + (0.5 − 0.866i)19-s + (0.680 − 0.733i)22-s + (−0.294 − 0.955i)23-s + (−0.733 − 0.680i)26-s + (0.733 − 0.680i)29-s + 31-s + (0.974 − 0.222i)32-s + (0.988 + 0.149i)34-s + ⋯
L(s)  = 1  + (−0.781 + 0.623i)2-s + (0.222 − 0.974i)4-s + (0.433 + 0.900i)8-s + (−0.988 + 0.149i)11-s + (0.149 + 0.988i)13-s + (−0.900 − 0.433i)16-s + (−0.680 − 0.733i)17-s + (0.5 − 0.866i)19-s + (0.680 − 0.733i)22-s + (−0.294 − 0.955i)23-s + (−0.733 − 0.680i)26-s + (0.733 − 0.680i)29-s + 31-s + (0.974 − 0.222i)32-s + (0.988 + 0.149i)34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.684 - 0.729i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.684 - 0.729i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2205\)    =    \(3^{2} \cdot 5 \cdot 7^{2}\)
Sign: $-0.684 - 0.729i$
Analytic conductor: \(236.960\)
Root analytic conductor: \(236.960\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2205} (277, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2205,\ (1:\ ),\ -0.684 - 0.729i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1095148882 - 0.2529557745i\)
\(L(\frac12)\) \(\approx\) \(0.1095148882 - 0.2529557745i\)
\(L(1)\) \(\approx\) \(0.6334515230 + 0.1051046710i\)
\(L(1)\) \(\approx\) \(0.6334515230 + 0.1051046710i\)

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 \( 1 + (-0.988 + 0.149i)T \)
13 \( 1 + (0.149 + 0.988i)T \)
17 \( 1 + (-0.680 - 0.733i)T \)
19 \( 1 + (0.5 - 0.866i)T \)
23 \( 1 + (-0.294 - 0.955i)T \)
29 \( 1 + (0.733 - 0.680i)T \)
31 \( 1 + T \)
37 \( 1 + (-0.294 + 0.955i)T \)
41 \( 1 + (0.0747 - 0.997i)T \)
43 \( 1 + (0.997 - 0.0747i)T \)
47 \( 1 + (-0.781 + 0.623i)T \)
53 \( 1 + (-0.294 - 0.955i)T \)
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 \( 1 + (-0.149 + 0.988i)T \)
79 \( 1 - T \)
83 \( 1 + (-0.149 + 0.988i)T \)
89 \( 1 + (-0.365 - 0.930i)T \)
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

−19.75494377705412808397754683, −19.193352635912091459985859619968, −18.22246266016578456313397543313, −17.87046139746526209489238549581, −17.20669635199426440187851057490, −16.14175701675577321958423579204, −15.79745428423284485073176829038, −14.91756438926104724461145833697, −13.74487222629253341284441064862, −13.115884031147169615232241935590, −12.45231773232572578964368394206, −11.71459006383024130826958936189, −10.73666139030775607564999521280, −10.411881587033200620074386375284, −9.60754535764029315246479343727, −8.68394866426852807445130909327, −7.98106490241150535086794923818, −7.52954758393058430854788301874, −6.38101501625949250700203991924, −5.539082225479650987991452794326, −4.49450468864551383270565474665, −3.472934832159184084894153420164, −2.86174273226841285319113520721, −1.87982801238452602287914403647, −0.9611305702009935982278649919, 0.07914672172190639154192413580, 0.949432624251373661508067686519, 2.19275764465697673949897693496, 2.76800140390302237687256265065, 4.40745516428119040561087936393, 4.89470453155098885807597909991, 5.89790736698262258781039706550, 6.76146419457212341606688294796, 7.22610393607221361273154189177, 8.27307494348900753474736738487, 8.73791347699038970441862456980, 9.70684529322891341839651821031, 10.190095069147150698484754627548, 11.20062698858769342224790321203, 11.652729075113406643903148996288, 12.8268058879946903912892564360, 13.780917155591641333721259621725, 14.156712338568677921271120035917, 15.2411510270991704237566755782, 15.895023695359541715045355347799, 16.17833339770611718535538929161, 17.3417514581917015958955554275, 17.702476744489947314199141502998, 18.594369574234475805772226644869, 19.01654154541651578454600187064

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