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Properties

Label 1-2404-2404.1011-r0-0-0
Degree $1$
Conductor $2404$
Sign $0.135 - 0.990i$
Analytic cond. $11.1641$
Root an. cond. $11.1641$
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  + (−0.0627 − 0.998i)3-s + i·5-s + (−0.917 + 0.397i)7-s + (−0.992 + 0.125i)9-s + (−0.278 + 0.960i)11-s + (−0.951 + 0.309i)13-s + (0.998 − 0.0627i)15-s + (−0.987 + 0.156i)17-s + (0.509 − 0.860i)19-s + (0.453 + 0.891i)21-s + (0.684 − 0.728i)23-s − 25-s + (0.187 + 0.982i)27-s + (−0.860 − 0.509i)29-s + (−0.0314 − 0.999i)31-s + ⋯
L(s)  = 1  + (−0.0627 − 0.998i)3-s + i·5-s + (−0.917 + 0.397i)7-s + (−0.992 + 0.125i)9-s + (−0.278 + 0.960i)11-s + (−0.951 + 0.309i)13-s + (0.998 − 0.0627i)15-s + (−0.987 + 0.156i)17-s + (0.509 − 0.860i)19-s + (0.453 + 0.891i)21-s + (0.684 − 0.728i)23-s − 25-s + (0.187 + 0.982i)27-s + (−0.860 − 0.509i)29-s + (−0.0314 − 0.999i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2404 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.135 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2404 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.135 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2404\)    =    \(2^{2} \cdot 601\)
Sign: $0.135 - 0.990i$
Analytic conductor: \(11.1641\)
Root analytic conductor: \(11.1641\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2404} (1011, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2404,\ (0:\ ),\ 0.135 - 0.990i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4227560351 - 0.3687505941i\)
\(L(\frac12)\) \(\approx\) \(0.4227560351 - 0.3687505941i\)
\(L(1)\) \(\approx\) \(0.7140952331 - 0.05493068313i\)
\(L(1)\) \(\approx\) \(0.7140952331 - 0.05493068313i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
601 \( 1 \)
good3 \( 1 + (-0.0627 - 0.998i)T \)
5 \( 1 + iT \)
7 \( 1 + (-0.917 + 0.397i)T \)
11 \( 1 + (-0.278 + 0.960i)T \)
13 \( 1 + (-0.951 + 0.309i)T \)
17 \( 1 + (-0.987 + 0.156i)T \)
19 \( 1 + (0.509 - 0.860i)T \)
23 \( 1 + (0.684 - 0.728i)T \)
29 \( 1 + (-0.860 - 0.509i)T \)
31 \( 1 + (-0.0314 - 0.999i)T \)
37 \( 1 + (-0.309 + 0.951i)T \)
41 \( 1 + (-0.661 + 0.750i)T \)
43 \( 1 + (-0.612 - 0.790i)T \)
47 \( 1 + (0.187 + 0.982i)T \)
53 \( 1 + (0.790 + 0.612i)T \)
59 \( 1 + (-0.707 - 0.707i)T \)
61 \( 1 + (-0.368 + 0.929i)T \)
67 \( 1 + (0.929 - 0.368i)T \)
71 \( 1 + (-0.156 + 0.987i)T \)
73 \( 1 + (0.218 + 0.975i)T \)
79 \( 1 + (0.995 - 0.0941i)T \)
83 \( 1 + (0.562 + 0.827i)T \)
89 \( 1 + (0.187 - 0.982i)T \)
97 \( 1 + (-0.453 - 0.891i)T \)
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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

−19.8085534127836733376146356896, −19.33628242337914232687334213446, −18.16993182089355509625816931988, −17.24173518226038991099256075650, −16.705867240259889308225257658364, −16.19211144501553270437306928027, −15.63820422877432323799798095737, −14.83918564155267195851807986309, −13.83117422551655059126043267348, −13.3212598113782352362036461635, −12.456238311745926623029030740043, −11.75537812055292878076696189719, −10.80780675276237700496807455005, −10.22606263138615424268389543598, −9.32473667443112348862432661013, −9.01811185977340525853640572517, −8.08532935170615908793363989136, −7.1658478255070153028617681071, −6.09732666350501227757861141107, −5.287433631898391652486473073282, −4.87479210370465459329466847215, −3.63600389024080403497595973030, −3.39078863559728932364252421232, −2.13187823233411399273097190417, −0.71000377213463058475421141036, 0.25357866972455727062089490393, 1.896646123192338007234609553315, 2.52614628713909993058678558933, 3.03611245431665940016200904760, 4.31324636578855984503263956748, 5.31040795463570104682790007428, 6.253378346642521088474490737642, 6.93866533637246156533495847261, 7.18226033301149495351984112876, 8.17842174552472558576098806195, 9.243802536331678851483224794993, 9.79023305401729445161096500381, 10.74481510375734140346583908611, 11.548399942905157713184655106664, 12.13923101190282426833456663792, 13.01407554980541165786956373378, 13.406728043073766049160816538045, 14.33494624962505381553484856156, 15.23284069385417603222816429528, 15.412495480977599440211814289328, 16.84687778836648335423218131869, 17.26062412459582160061679647132, 18.20138466600261500460083506967, 18.59611354097227542218224262818, 19.26782425526460483918159114149

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