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Properties

Label 1-2205-2205.499-r0-0-0
Degree $1$
Conductor $2205$
Sign $0.311 + 0.950i$
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  + (0.900 + 0.433i)2-s + (0.623 + 0.781i)4-s + (0.222 + 0.974i)8-s + (0.0747 + 0.997i)11-s + (−0.0747 − 0.997i)13-s + (−0.222 + 0.974i)16-s + (−0.365 − 0.930i)17-s + (−0.5 − 0.866i)19-s + (−0.365 + 0.930i)22-s + (0.988 − 0.149i)23-s + (0.365 − 0.930i)26-s + (0.365 + 0.930i)29-s + 31-s + (−0.623 + 0.781i)32-s + (0.0747 − 0.997i)34-s + ⋯
L(s)  = 1  + (0.900 + 0.433i)2-s + (0.623 + 0.781i)4-s + (0.222 + 0.974i)8-s + (0.0747 + 0.997i)11-s + (−0.0747 − 0.997i)13-s + (−0.222 + 0.974i)16-s + (−0.365 − 0.930i)17-s + (−0.5 − 0.866i)19-s + (−0.365 + 0.930i)22-s + (0.988 − 0.149i)23-s + (0.365 − 0.930i)26-s + (0.365 + 0.930i)29-s + 31-s + (−0.623 + 0.781i)32-s + (0.0747 − 0.997i)34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.311 + 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.311 + 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2205\)    =    \(3^{2} \cdot 5 \cdot 7^{2}\)
Sign: $0.311 + 0.950i$
Analytic conductor: \(10.2399\)
Root analytic conductor: \(10.2399\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2205} (499, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2205,\ (0:\ ),\ 0.311 + 0.950i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.432430871 + 1.761974940i\)
\(L(\frac12)\) \(\approx\) \(2.432430871 + 1.761974940i\)
\(L(1)\) \(\approx\) \(1.723918164 + 0.6589502371i\)
\(L(1)\) \(\approx\) \(1.723918164 + 0.6589502371i\)

Euler product

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

−19.34735068622770008006424029217, −19.23816244089676019261045622847, −18.49291327658146984094858486968, −17.195449586152401830448610812102, −16.678597790396457526191125852191, −15.824896670133842150381412214797, −15.092336863865206852601450024798, −14.43532490307555143127050003570, −13.63651206213771468271026040150, −13.227211220320620908822308056129, −12.166178436183080653960130298626, −11.73853898226715589903224284353, −10.81491293267551192015213120004, −10.36134675943143918879617234396, −9.29560886307637052700084227655, −8.55362000395995748206769776549, −7.5323688367184345197670637179, −6.46521465687929985232536144014, −6.10341456856155435634627095885, −5.15075308797806105413128217356, −4.18322917157980039806912588897, −3.71045096792932791481690788057, −2.637567327278458499590096835630, −1.869398706266796087299678031943, −0.82456954559277952904851831216, 1.08810027464909288155291406683, 2.56844203567377472773349211225, 2.82173675449129074150830371611, 4.1084702699453920704712618037, 4.80062776472172014985812948253, 5.34863694048051100655775849861, 6.47716603896326027475807061765, 7.01018617885173983919543477855, 7.75150610150036836983699322166, 8.61667109990771073865594204681, 9.494479367644124982992533229756, 10.511521994692634609233602272010, 11.22093801756426358547456754851, 12.038848272325835977272215194633, 12.77595151356679202717256004468, 13.26372200155074327248029409001, 14.08944872610167609813027000302, 15.02610484248182634080808157468, 15.26996580150437484000393650979, 16.10473384225913522949827580672, 16.931134164268383842009279241796, 17.646760142403747303019459323768, 18.13293173193120213019250680924, 19.3806573716492938541834767676, 20.15901350410619691396610571380

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