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Properties

Label 1-2205-2205.907-r1-0-0
Degree $1$
Conductor $2205$
Sign $-0.804 + 0.594i$
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  + (0.433 − 0.900i)2-s + (−0.623 − 0.781i)4-s + (−0.974 + 0.222i)8-s + (0.826 − 0.563i)11-s + (−0.563 − 0.826i)13-s + (−0.222 + 0.974i)16-s + (0.149 − 0.988i)17-s + (0.5 − 0.866i)19-s + (−0.149 − 0.988i)22-s + (−0.930 − 0.365i)23-s + (−0.988 + 0.149i)26-s + (0.988 + 0.149i)29-s + 31-s + (0.781 + 0.623i)32-s + (−0.826 − 0.563i)34-s + ⋯
L(s)  = 1  + (0.433 − 0.900i)2-s + (−0.623 − 0.781i)4-s + (−0.974 + 0.222i)8-s + (0.826 − 0.563i)11-s + (−0.563 − 0.826i)13-s + (−0.222 + 0.974i)16-s + (0.149 − 0.988i)17-s + (0.5 − 0.866i)19-s + (−0.149 − 0.988i)22-s + (−0.930 − 0.365i)23-s + (−0.988 + 0.149i)26-s + (0.988 + 0.149i)29-s + 31-s + (0.781 + 0.623i)32-s + (−0.826 − 0.563i)34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.804 + 0.594i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.804 + 0.594i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2205\)    =    \(3^{2} \cdot 5 \cdot 7^{2}\)
Sign: $-0.804 + 0.594i$
Analytic conductor: \(236.960\)
Root analytic conductor: \(236.960\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2205} (907, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2205,\ (1:\ ),\ -0.804 + 0.594i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.6724350998 - 2.042471863i\)
\(L(\frac12)\) \(\approx\) \(-0.6724350998 - 2.042471863i\)
\(L(1)\) \(\approx\) \(0.8526009866 - 0.8874318406i\)
\(L(1)\) \(\approx\) \(0.8526009866 - 0.8874318406i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good2 \( 1 + (0.433 - 0.900i)T \)
11 \( 1 + (0.826 - 0.563i)T \)
13 \( 1 + (-0.563 - 0.826i)T \)
17 \( 1 + (0.149 - 0.988i)T \)
19 \( 1 + (0.5 - 0.866i)T \)
23 \( 1 + (-0.930 - 0.365i)T \)
29 \( 1 + (0.988 + 0.149i)T \)
31 \( 1 + T \)
37 \( 1 + (-0.930 + 0.365i)T \)
41 \( 1 + (0.955 + 0.294i)T \)
43 \( 1 + (-0.294 - 0.955i)T \)
47 \( 1 + (0.433 - 0.900i)T \)
53 \( 1 + (-0.930 - 0.365i)T \)
59 \( 1 + (0.222 - 0.974i)T \)
61 \( 1 + (0.623 - 0.781i)T \)
67 \( 1 + iT \)
71 \( 1 + (0.623 + 0.781i)T \)
73 \( 1 + (0.563 - 0.826i)T \)
79 \( 1 - T \)
83 \( 1 + (0.563 - 0.826i)T \)
89 \( 1 + (-0.0747 + 0.997i)T \)
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

−19.844577324397125356086810470691, −19.29401430820907821964952908965, −18.378475594066951635916626846148, −17.48305060720338705132242772128, −17.17447166206635491968061932423, −16.283206875018263834352529508144, −15.70168400098696363203212603694, −14.83063138101560257760596962988, −14.200626849076340434315270717308, −13.82547321759584369266887033447, −12.58256710229807710493117446745, −12.22277694118845955778320256881, −11.509490833723066114766131880937, −10.17701427336338458234945381358, −9.57383162323629245761300491804, −8.74868739964670790146054586710, −7.92938956945241071023346220339, −7.27120551524589735167363103930, −6.36860233957819120512192827625, −5.919177135840466413290611510490, −4.76759106697661453763082325414, −4.18965521073072103246383926455, −3.44341867225763832230540407543, −2.25428304583352423527155557594, −1.1804711663275962659068705325, 0.38011911101970792065467016932, 0.9377348110186896100176839095, 2.17942113414860885962331601141, 2.95219320999547833441845404506, 3.64633840615192718147366512993, 4.6911690095256089085936944931, 5.24015375253642765052455728713, 6.200643601063904440199946821743, 6.98697571157287662742954205374, 8.16208125621145243019562472071, 8.88792158216636433170783997507, 9.75258667256538942620106402291, 10.26076921301984324228333596724, 11.22181438012406536832409301896, 11.84493759175035640406931246927, 12.37201379263251798431546149520, 13.31949006449704412097646176854, 14.01430375094474625034307323484, 14.42311255856410131899373186846, 15.498530142039265322659509810754, 16.015208660001560805151885961590, 17.26431560997636047621315571096, 17.718947519722117044506413091583, 18.592969689355039477603949295447, 19.266810772592077894488315815373

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