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Properties

Label 1-4725-4725.2027-r0-0-0
Degree $1$
Conductor $4725$
Sign $-0.503 + 0.863i$
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  + (−0.927 + 0.374i)2-s + (0.719 − 0.694i)4-s + (−0.406 + 0.913i)8-s + (0.374 + 0.927i)11-s + (−0.788 + 0.615i)13-s + (0.0348 − 0.999i)16-s + (0.587 + 0.809i)17-s + (0.809 − 0.587i)19-s + (−0.694 − 0.719i)22-s + (−0.999 + 0.0348i)23-s + (0.5 − 0.866i)26-s + (0.559 + 0.829i)29-s + (0.438 + 0.898i)31-s + (0.342 + 0.939i)32-s + (−0.848 − 0.529i)34-s + ⋯
L(s)  = 1  + (−0.927 + 0.374i)2-s + (0.719 − 0.694i)4-s + (−0.406 + 0.913i)8-s + (0.374 + 0.927i)11-s + (−0.788 + 0.615i)13-s + (0.0348 − 0.999i)16-s + (0.587 + 0.809i)17-s + (0.809 − 0.587i)19-s + (−0.694 − 0.719i)22-s + (−0.999 + 0.0348i)23-s + (0.5 − 0.866i)26-s + (0.559 + 0.829i)29-s + (0.438 + 0.898i)31-s + (0.342 + 0.939i)32-s + (−0.848 − 0.529i)34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4725 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.503 + 0.863i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4725 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.503 + 0.863i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4725\)    =    \(3^{3} \cdot 5^{2} \cdot 7\)
Sign: $-0.503 + 0.863i$
Analytic conductor: \(21.9428\)
Root analytic conductor: \(21.9428\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4725} (2027, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4725,\ (0:\ ),\ -0.503 + 0.863i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4976055340 + 0.8659838666i\)
\(L(\frac12)\) \(\approx\) \(0.4976055340 + 0.8659838666i\)
\(L(1)\) \(\approx\) \(0.6859592169 + 0.2586333772i\)
\(L(1)\) \(\approx\) \(0.6859592169 + 0.2586333772i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good2 \( 1 + (-0.927 + 0.374i)T \)
11 \( 1 + (0.374 + 0.927i)T \)
13 \( 1 + (-0.788 + 0.615i)T \)
17 \( 1 + (0.587 + 0.809i)T \)
19 \( 1 + (0.809 - 0.587i)T \)
23 \( 1 + (-0.999 + 0.0348i)T \)
29 \( 1 + (0.559 + 0.829i)T \)
31 \( 1 + (0.438 + 0.898i)T \)
37 \( 1 + (0.743 + 0.669i)T \)
41 \( 1 + (0.615 + 0.788i)T \)
43 \( 1 + (0.984 - 0.173i)T \)
47 \( 1 + (-0.898 - 0.438i)T \)
53 \( 1 + (-0.994 - 0.104i)T \)
59 \( 1 + (-0.615 - 0.788i)T \)
61 \( 1 + (0.990 + 0.139i)T \)
67 \( 1 + (0.0697 - 0.997i)T \)
71 \( 1 + (-0.913 + 0.406i)T \)
73 \( 1 + (0.743 - 0.669i)T \)
79 \( 1 + (0.997 - 0.0697i)T \)
83 \( 1 + (0.275 + 0.961i)T \)
89 \( 1 + (0.309 - 0.951i)T \)
97 \( 1 + (0.898 + 0.438i)T \)
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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

−17.88046792944351112482660227422, −17.45608949577351823069270403648, −16.58973784697603368627472655685, −16.14654042887942352164229468140, −15.55443897864065501155055273125, −14.52685456380338522214097452239, −13.98077003232678426379611398956, −13.09402778522693338284362804214, −12.250853566715165946875790443555, −11.807569069800106889134083135461, −11.160043637792476396733362602041, −10.34853650847567640012584878207, −9.66992578759978821190804227370, −9.30935688578328776935369895849, −8.20172242535976129121215157354, −7.83286678547834451677698312315, −7.188721579777597441381079608331, −6.10021576426241392873516935411, −5.70110172705451445336437558098, −4.4718531161055601810382743067, −3.62712967716622834608521302657, −2.87331043752872527121693039708, −2.26040223976944577162253829233, −1.10323471556193526632150707097, −0.45011609169512984599923897910, 1.02571127958130782932170139016, 1.7553421095160303133853338108, 2.5296512527005019364838022340, 3.4850235222513438582746370355, 4.65362620602337194932873019911, 5.12348894409201378977460819369, 6.275472080806635176935754062612, 6.647125520545022047060390290897, 7.554084319208713670866590827300, 7.95833123141470149818548371616, 8.91425061998972194754410767318, 9.56970151502759748131880486963, 10.00444923998859763815253155626, 10.74141303648038601259572051742, 11.65383469304963720193561142716, 12.11246520743324862094434317610, 12.84867119731775387582978945087, 14.104692885608999863337046328991, 14.40625609016467840401199591617, 15.12812446308932386763351404177, 15.868027294476846135541943662258, 16.430823582233204150656202892017, 17.14798646912183331847049851600, 17.71445024296193694579645984659, 18.19107443991283597836313118418

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