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Properties

Label 1-2205-2205.59-r0-0-0
Degree $1$
Conductor $2205$
Sign $0.992 + 0.120i$
Analytic cond. $10.2399$
Root an. cond. $10.2399$
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  + (−0.733 + 0.680i)2-s + (0.0747 − 0.997i)4-s + (0.623 + 0.781i)8-s + (0.222 + 0.974i)11-s + (−0.733 + 0.680i)13-s + (−0.988 − 0.149i)16-s + (−0.0747 − 0.997i)17-s + (0.5 − 0.866i)19-s + (−0.826 − 0.563i)22-s + (−0.900 + 0.433i)23-s + (0.0747 − 0.997i)26-s + (−0.826 + 0.563i)29-s + (0.5 − 0.866i)31-s + (0.826 − 0.563i)32-s + (0.733 + 0.680i)34-s + ⋯
L(s)  = 1  + (−0.733 + 0.680i)2-s + (0.0747 − 0.997i)4-s + (0.623 + 0.781i)8-s + (0.222 + 0.974i)11-s + (−0.733 + 0.680i)13-s + (−0.988 − 0.149i)16-s + (−0.0747 − 0.997i)17-s + (0.5 − 0.866i)19-s + (−0.826 − 0.563i)22-s + (−0.900 + 0.433i)23-s + (0.0747 − 0.997i)26-s + (−0.826 + 0.563i)29-s + (0.5 − 0.866i)31-s + (0.826 − 0.563i)32-s + (0.733 + 0.680i)34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.992 + 0.120i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.992 + 0.120i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2205\)    =    \(3^{2} \cdot 5 \cdot 7^{2}\)
Sign: $0.992 + 0.120i$
Analytic conductor: \(10.2399\)
Root analytic conductor: \(10.2399\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2205} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2205,\ (0:\ ),\ 0.992 + 0.120i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8909697874 + 0.05401625016i\)
\(L(\frac12)\) \(\approx\) \(0.8909697874 + 0.05401625016i\)
\(L(1)\) \(\approx\) \(0.6982148694 + 0.1635785773i\)
\(L(1)\) \(\approx\) \(0.6982148694 + 0.1635785773i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good2 \( 1 + (-0.733 + 0.680i)T \)
11 \( 1 + (0.222 + 0.974i)T \)
13 \( 1 + (-0.733 + 0.680i)T \)
17 \( 1 + (-0.0747 - 0.997i)T \)
19 \( 1 + (0.5 - 0.866i)T \)
23 \( 1 + (-0.900 + 0.433i)T \)
29 \( 1 + (-0.826 + 0.563i)T \)
31 \( 1 + (0.5 - 0.866i)T \)
37 \( 1 + (-0.826 + 0.563i)T \)
41 \( 1 + (0.365 - 0.930i)T \)
43 \( 1 + (-0.365 - 0.930i)T \)
47 \( 1 + (0.733 - 0.680i)T \)
53 \( 1 + (0.826 + 0.563i)T \)
59 \( 1 + (0.365 + 0.930i)T \)
61 \( 1 + (-0.0747 - 0.997i)T \)
67 \( 1 + (0.5 - 0.866i)T \)
71 \( 1 + (0.900 - 0.433i)T \)
73 \( 1 + (0.955 - 0.294i)T \)
79 \( 1 + (-0.5 - 0.866i)T \)
83 \( 1 + (0.733 + 0.680i)T \)
89 \( 1 + (-0.733 - 0.680i)T \)
97 \( 1 + (-0.5 + 0.866i)T \)
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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

−19.58383454020441030084827419761, −19.15125985782815901409081430951, −18.30320823293216858537866374647, −17.67490046577912926264915271263, −16.933426722921835658785398077327, −16.34786421867521736261535218447, −15.59550682213176786574732708625, −14.56585737380771705880938145784, −13.85066280942140471552513749816, −12.885242774911686124329204190906, −12.37124766095809330752259423960, −11.57618248816378381312364832802, −10.8513268229067208118662399176, −10.11255416161161918771642654027, −9.5824610697661965553152309553, −8.43442298903444261238608797613, −8.16634470306517640546668144220, −7.24899703785749627982750763755, −6.25035975421646662016805345799, −5.44189593482990859493359314392, −4.199686445232161688349861818166, −3.52750506662509326910003778289, −2.691784739607280696829997871912, −1.75394381946291026402832360118, −0.78072253023386798688695871347, 0.52579050580925854755465438734, 1.83508449838575875370245473279, 2.42473979881370800530009799385, 3.88902057707906556577798560260, 4.86101416213690687205881224963, 5.38377796206220641972214141094, 6.522218305121143158437189156795, 7.20777208979509717752751028572, 7.57922167350439443098375293840, 8.75073124264777248242737293626, 9.41860942842992714942027630691, 9.856802801348776774028208796097, 10.78132584917058434011911932845, 11.71530451540312234081764617735, 12.213003913140840755917595414857, 13.597480129215849391376450262666, 13.95443493962872919283404188074, 14.94589852358097744089090985072, 15.44072129714124464371145487304, 16.16762601483175009474088818815, 17.04025705239282713117166864099, 17.42848385733467978564832696698, 18.32458526460680105058683783647, 18.77172681315445345183482951561, 19.849164759876052407301876835979

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