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Properties

Label 1-2205-2205.1247-r0-0-0
Degree $1$
Conductor $2205$
Sign $-0.686 - 0.726i$
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  + (0.930 − 0.365i)2-s + (0.733 − 0.680i)4-s + (0.433 − 0.900i)8-s + (−0.365 − 0.930i)11-s + (−0.149 + 0.988i)13-s + (0.0747 − 0.997i)16-s + (0.974 + 0.222i)17-s − 19-s + (−0.680 − 0.733i)22-s + (−0.680 − 0.733i)23-s + (0.222 + 0.974i)26-s + (−0.733 − 0.680i)29-s + (−0.5 − 0.866i)31-s + (−0.294 − 0.955i)32-s + (0.988 − 0.149i)34-s + ⋯
L(s)  = 1  + (0.930 − 0.365i)2-s + (0.733 − 0.680i)4-s + (0.433 − 0.900i)8-s + (−0.365 − 0.930i)11-s + (−0.149 + 0.988i)13-s + (0.0747 − 0.997i)16-s + (0.974 + 0.222i)17-s − 19-s + (−0.680 − 0.733i)22-s + (−0.680 − 0.733i)23-s + (0.222 + 0.974i)26-s + (−0.733 − 0.680i)29-s + (−0.5 − 0.866i)31-s + (−0.294 − 0.955i)32-s + (0.988 − 0.149i)34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.686 - 0.726i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.686 - 0.726i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2205\)    =    \(3^{2} \cdot 5 \cdot 7^{2}\)
Sign: $-0.686 - 0.726i$
Analytic conductor: \(10.2399\)
Root analytic conductor: \(10.2399\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2205} (1247, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2205,\ (0:\ ),\ -0.686 - 0.726i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8875228837 - 2.060041304i\)
\(L(\frac12)\) \(\approx\) \(0.8875228837 - 2.060041304i\)
\(L(1)\) \(\approx\) \(1.464665961 - 0.7252942314i\)
\(L(1)\) \(\approx\) \(1.464665961 - 0.7252942314i\)

Euler product

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

−20.14875099418279866941252404921, −19.56417180286705767608242803220, −18.391634205969766274076626808006, −17.73190200193019687992973243565, −16.97631048262367668715331130619, −16.31209915323598850373851597857, −15.39482974902598191884847138785, −15.03176122748836880681215249607, −14.25876052801872561360882921167, −13.47098543639227675225305205107, −12.598971902552937142876764986586, −12.37554728075175120119827077982, −11.36909798435288616176120136154, −10.49454905655350999327031288198, −9.88774656867369512817031332791, −8.66674477242924935626009929216, −7.85384217382972865048407696298, −7.282690658550701751007189291, −6.47220085192024864497829511176, −5.41599381641310048279939367437, −5.12258785693380446211103706288, −4.007673537588027510977535303855, −3.30578657132925639635219443544, −2.39930459754136592844745961972, −1.4897962957513105755984757263, 0.481572545800960849129447796611, 1.85042340493475892850688432719, 2.404473523932685123219950793, 3.62341519740154046097209457958, 4.01283810058009001915026515109, 5.09935538557170846058207529367, 5.811235413272425306467933974491, 6.4778129743686800394946620508, 7.36215629070980177498357040743, 8.29157400285344258411014446565, 9.20148354670081500562667911163, 10.18856856718977885779719412634, 10.72011131886842766332357557350, 11.58413362457822209396777621396, 12.14902901574897484353801217375, 12.949137839757590304954765129725, 13.692386573803642687788748784573, 14.26082426562559587931692143439, 14.97084043451264708724530646199, 15.71661917334431962832389782011, 16.64933475946471975667934053560, 16.88963989513764332187021706388, 18.3886447460628397277089857334, 18.94966120388117167000738886959, 19.38000496360428370788979396886

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