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Properties

Label 1-4725-4725.1208-r0-0-0
Degree $1$
Conductor $4725$
Sign $-0.842 - 0.539i$
Analytic cond. $21.9428$
Root an. cond. $21.9428$
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  + (0.694 − 0.719i)2-s + (−0.0348 − 0.999i)4-s + (−0.743 − 0.669i)8-s + (0.719 + 0.694i)11-s + (0.970 − 0.241i)13-s + (−0.997 + 0.0697i)16-s + (−0.951 + 0.309i)17-s + (−0.309 − 0.951i)19-s + (0.999 − 0.0348i)22-s + (−0.0697 + 0.997i)23-s + (0.5 − 0.866i)26-s + (−0.374 − 0.927i)29-s + (−0.615 − 0.788i)31-s + (−0.642 + 0.766i)32-s + (−0.438 + 0.898i)34-s + ⋯
L(s)  = 1  + (0.694 − 0.719i)2-s + (−0.0348 − 0.999i)4-s + (−0.743 − 0.669i)8-s + (0.719 + 0.694i)11-s + (0.970 − 0.241i)13-s + (−0.997 + 0.0697i)16-s + (−0.951 + 0.309i)17-s + (−0.309 − 0.951i)19-s + (0.999 − 0.0348i)22-s + (−0.0697 + 0.997i)23-s + (0.5 − 0.866i)26-s + (−0.374 − 0.927i)29-s + (−0.615 − 0.788i)31-s + (−0.642 + 0.766i)32-s + (−0.438 + 0.898i)34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4725 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.842 - 0.539i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4725 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.842 - 0.539i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4725\)    =    \(3^{3} \cdot 5^{2} \cdot 7\)
Sign: $-0.842 - 0.539i$
Analytic conductor: \(21.9428\)
Root analytic conductor: \(21.9428\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4725} (1208, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4725,\ (0:\ ),\ -0.842 - 0.539i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6035055248 - 2.061454688i\)
\(L(\frac12)\) \(\approx\) \(0.6035055248 - 2.061454688i\)
\(L(1)\) \(\approx\) \(1.211640182 - 0.7918634710i\)
\(L(1)\) \(\approx\) \(1.211640182 - 0.7918634710i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good2 \( 1 + (0.694 - 0.719i)T \)
11 \( 1 + (0.719 + 0.694i)T \)
13 \( 1 + (0.970 - 0.241i)T \)
17 \( 1 + (-0.951 + 0.309i)T \)
19 \( 1 + (-0.309 - 0.951i)T \)
23 \( 1 + (-0.0697 + 0.997i)T \)
29 \( 1 + (-0.374 - 0.927i)T \)
31 \( 1 + (-0.615 - 0.788i)T \)
37 \( 1 + (0.994 + 0.104i)T \)
41 \( 1 + (0.241 + 0.970i)T \)
43 \( 1 + (0.342 - 0.939i)T \)
47 \( 1 + (-0.788 - 0.615i)T \)
53 \( 1 + (0.207 + 0.978i)T \)
59 \( 1 + (-0.241 - 0.970i)T \)
61 \( 1 + (0.961 - 0.275i)T \)
67 \( 1 + (-0.139 - 0.990i)T \)
71 \( 1 + (-0.669 - 0.743i)T \)
73 \( 1 + (0.994 - 0.104i)T \)
79 \( 1 + (-0.990 - 0.139i)T \)
83 \( 1 + (0.529 - 0.848i)T \)
89 \( 1 + (-0.809 + 0.587i)T \)
97 \( 1 + (0.788 + 0.615i)T \)
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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

−18.21153526440478761155015544767, −17.77199270335325549766244770288, −16.78703617600394626392352841418, −16.29853614408366261630211505969, −15.95756300695022622584594435544, −14.94531870215686076436674471784, −14.33807830675435839119485033771, −13.9914784038633479093846274332, −12.956048646546672562624268751671, −12.749886172085634105686043165231, −11.64584851464896974772560110979, −11.242704992400191688302227111983, −10.44573250988290587514666402140, −9.23952534069268830225842297681, −8.71155771536871615136132800899, −8.21523549895828288935780102124, −7.21495173118917621353595321043, −6.55874990032350391590085572544, −6.04728052696978699293467465783, −5.32613571106156645175523957615, −4.333650573723111629218630078918, −3.878519451816349449087666348469, −3.10474873561532461655299879472, −2.17511417775551103040639043606, −1.106792141033478808744355568811, 0.45702882177396308734347368682, 1.559801922586209657279944136552, 2.12035922590171955725206766798, 3.046736128800724086504484422632, 3.952866585155017103503130793161, 4.31648061829360508672036594650, 5.22114324859171987208525201496, 6.08139249813138822517518166912, 6.55064675379228616260340090585, 7.43744922449993434682797686111, 8.445207846273041589559045482446, 9.339340320018797335733177245879, 9.62185526514897871449192347361, 10.69568267829162107602964190028, 11.23136301245059545443164957676, 11.68797543734392813228568650231, 12.58222158542875742452648898074, 13.31274413359948555616773409162, 13.51601411391550746741041817605, 14.52325017389190854963364726501, 15.2734516804334731091484340923, 15.44667393192872316571916091373, 16.48867334534934870450490227854, 17.427377965238707389860287357045, 17.9116827571918391228926276343

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