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Properties

Label 1-2205-2205.88-r1-0-0
Degree $1$
Conductor $2205$
Sign $-0.964 - 0.262i$
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  + (−0.974 + 0.222i)2-s + (0.900 − 0.433i)4-s + (−0.781 + 0.623i)8-s + (0.955 + 0.294i)11-s + (−0.294 + 0.955i)13-s + (0.623 − 0.781i)16-s + (0.997 − 0.0747i)17-s + (0.5 − 0.866i)19-s + (−0.997 − 0.0747i)22-s + (−0.563 + 0.826i)23-s + (0.0747 − 0.997i)26-s + (−0.0747 − 0.997i)29-s + 31-s + (−0.433 + 0.900i)32-s + (−0.955 + 0.294i)34-s + ⋯
L(s)  = 1  + (−0.974 + 0.222i)2-s + (0.900 − 0.433i)4-s + (−0.781 + 0.623i)8-s + (0.955 + 0.294i)11-s + (−0.294 + 0.955i)13-s + (0.623 − 0.781i)16-s + (0.997 − 0.0747i)17-s + (0.5 − 0.866i)19-s + (−0.997 − 0.0747i)22-s + (−0.563 + 0.826i)23-s + (0.0747 − 0.997i)26-s + (−0.0747 − 0.997i)29-s + 31-s + (−0.433 + 0.900i)32-s + (−0.955 + 0.294i)34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.964 - 0.262i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.964 - 0.262i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2205\)    =    \(3^{2} \cdot 5 \cdot 7^{2}\)
Sign: $-0.964 - 0.262i$
Analytic conductor: \(236.960\)
Root analytic conductor: \(236.960\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2205} (88, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2205,\ (1:\ ),\ -0.964 - 0.262i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.01122194745 + 0.08398549298i\)
\(L(\frac12)\) \(\approx\) \(0.01122194745 + 0.08398549298i\)
\(L(1)\) \(\approx\) \(0.6728684660 + 0.08688212062i\)
\(L(1)\) \(\approx\) \(0.6728684660 + 0.08688212062i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good2 \( 1 + (-0.974 + 0.222i)T \)
11 \( 1 + (0.955 + 0.294i)T \)
13 \( 1 + (-0.294 + 0.955i)T \)
17 \( 1 + (0.997 - 0.0747i)T \)
19 \( 1 + (0.5 - 0.866i)T \)
23 \( 1 + (-0.563 + 0.826i)T \)
29 \( 1 + (-0.0747 - 0.997i)T \)
31 \( 1 + T \)
37 \( 1 + (-0.563 - 0.826i)T \)
41 \( 1 + (-0.988 + 0.149i)T \)
43 \( 1 + (0.149 - 0.988i)T \)
47 \( 1 + (-0.974 + 0.222i)T \)
53 \( 1 + (-0.563 + 0.826i)T \)
59 \( 1 + (-0.623 + 0.781i)T \)
61 \( 1 + (-0.900 - 0.433i)T \)
67 \( 1 - iT \)
71 \( 1 + (-0.900 + 0.433i)T \)
73 \( 1 + (0.294 + 0.955i)T \)
79 \( 1 - T \)
83 \( 1 + (0.294 + 0.955i)T \)
89 \( 1 + (0.733 + 0.680i)T \)
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

−19.1260425290004172504227367810, −18.461888213480890473434923032799, −17.78387046826616018419981992104, −17.01452179976224253752717140289, −16.48023008116303225146430674948, −15.80616526137865973450487011603, −14.80555491854218176700576211257, −14.32403129354323137292890868257, −13.17877942306625165286952956255, −12.18024950378247982469242112925, −11.96997286399598121989158517624, −10.94224269771453608284399026063, −10.092286509900324021343920603765, −9.78961602620204681913786617554, −8.65993594569497638974406471120, −8.15909808208131191291912945488, −7.399142900322620424113284662227, −6.469892009485983327295367589413, −5.84270999320437165953429486476, −4.72100703714354695866162787778, −3.40094006697987560036995305282, −3.09714570368689848611268790703, −1.727565019728476099779814524767, −1.092269203023171852797046953638, −0.02177192236026248787159409861, 1.13964954497216603872188823019, 1.84071627924822700620382933433, 2.85716650823352248928149717892, 3.87835485937350881906087163524, 4.925493410165479410993272307183, 5.882878187392959559781254271943, 6.654752546544250976210964229551, 7.317905476953658478430123391071, 8.03207750035371860328854622151, 9.01930119169574389686839608129, 9.52845814036749473828188650706, 10.11851533130078610267323437671, 11.16169849523130155936946653978, 11.86060659323532284460219520431, 12.2141184480974095488980632689, 13.71023540828644620299736762723, 14.17343825750446163243928383366, 15.10046718008090827146981015738, 15.692772018848317208087453121, 16.52349753659213623702504835166, 17.15747207600772166184855746996, 17.62834530441425340575690968151, 18.56264031533903153886069016950, 19.21511513757438430296836918577, 19.7029044037053605569697499144

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