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Properties

Label 1-15e2-225.178-r1-0-0
Degree $1$
Conductor $225$
Sign $0.0279 + 0.999i$
Analytic cond. $24.1796$
Root an. cond. $24.1796$
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  + (0.994 + 0.104i)2-s + (0.978 + 0.207i)4-s + (−0.866 + 0.5i)7-s + (0.951 + 0.309i)8-s + (−0.104 + 0.994i)11-s + (0.994 − 0.104i)13-s + (−0.913 + 0.406i)14-s + (0.913 + 0.406i)16-s + (−0.951 − 0.309i)17-s + (−0.309 + 0.951i)19-s + (−0.207 + 0.978i)22-s + (0.406 + 0.913i)23-s + 26-s + (−0.951 + 0.309i)28-s + (−0.669 + 0.743i)29-s + ⋯
L(s)  = 1  + (0.994 + 0.104i)2-s + (0.978 + 0.207i)4-s + (−0.866 + 0.5i)7-s + (0.951 + 0.309i)8-s + (−0.104 + 0.994i)11-s + (0.994 − 0.104i)13-s + (−0.913 + 0.406i)14-s + (0.913 + 0.406i)16-s + (−0.951 − 0.309i)17-s + (−0.309 + 0.951i)19-s + (−0.207 + 0.978i)22-s + (0.406 + 0.913i)23-s + 26-s + (−0.951 + 0.309i)28-s + (−0.669 + 0.743i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0279 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0279 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $0.0279 + 0.999i$
Analytic conductor: \(24.1796\)
Root analytic conductor: \(24.1796\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (178, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 225,\ (1:\ ),\ 0.0279 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.215881310 + 2.154850416i\)
\(L(\frac12)\) \(\approx\) \(2.215881310 + 2.154850416i\)
\(L(1)\) \(\approx\) \(1.742628858 + 0.5910852881i\)
\(L(1)\) \(\approx\) \(1.742628858 + 0.5910852881i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
good2 \( 1 + (0.994 + 0.104i)T \)
7 \( 1 + (-0.866 + 0.5i)T \)
11 \( 1 + (-0.104 + 0.994i)T \)
13 \( 1 + (0.994 - 0.104i)T \)
17 \( 1 + (-0.951 - 0.309i)T \)
19 \( 1 + (-0.309 + 0.951i)T \)
23 \( 1 + (0.406 + 0.913i)T \)
29 \( 1 + (-0.669 + 0.743i)T \)
31 \( 1 + (0.669 + 0.743i)T \)
37 \( 1 + (0.587 + 0.809i)T \)
41 \( 1 + (-0.104 - 0.994i)T \)
43 \( 1 + (0.866 - 0.5i)T \)
47 \( 1 + (-0.743 - 0.669i)T \)
53 \( 1 + (-0.951 + 0.309i)T \)
59 \( 1 + (0.104 + 0.994i)T \)
61 \( 1 + (-0.104 + 0.994i)T \)
67 \( 1 + (0.743 - 0.669i)T \)
71 \( 1 + (0.309 + 0.951i)T \)
73 \( 1 + (0.587 - 0.809i)T \)
79 \( 1 + (-0.669 + 0.743i)T \)
83 \( 1 + (-0.207 - 0.978i)T \)
89 \( 1 + (0.809 + 0.587i)T \)
97 \( 1 + (-0.743 - 0.669i)T \)
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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

−26.00612017230334342767883229983, −24.79081557518906504293456165123, −24.00425329810206769670631531772, −23.10675298132865132850398510586, −22.33852747518877944008463856297, −21.42985250657996921592046419198, −20.50552529049979074189919723700, −19.56102897924212749108318005574, −18.77839430945138647022549192715, −17.18610471784669344476193134061, −16.18590331173170428817080789590, −15.557583539923719090349583034761, −14.30744235881405612016844903919, −13.244527521486094265566791164612, −12.95644948584171582868792810753, −11.28387563432597168839351131692, −10.85284275607369120313440273673, −9.42572864530804616692606205599, −8.05257326402654922046052377659, −6.57717031318039533951514361720, −6.11557621685656785951712288261, −4.55771356369294396576743079975, −3.59025348695782509461144875137, −2.49546873271741408429446799304, −0.70991427404637047592662408691, 1.748657742968057613809893369851, 3.03285436644448518919472645619, 4.095032851787235632481995602627, 5.365012751204155209440382518, 6.36364692461998098073463038922, 7.27063203373809788761180908331, 8.69264772653147135317047184832, 9.99387152561582310112098879061, 11.11949357478429216768978632128, 12.23879952004965619444317834069, 12.97124930829433074419435550957, 13.84749098986576769962140786009, 15.148456819697125020645517676216, 15.68255238844606417867497190473, 16.66099073207546787615133350114, 17.915910497285476137082244694, 19.0995348588158906802658496978, 20.13217054898407327216977337305, 20.91023804438436677804641243485, 21.96333285014549852327331233100, 22.79899940009221253058058746431, 23.36656285534720188378777829383, 24.57207594307893526500911326586, 25.53958239369925157686896032047, 25.86939476689529973586607063323

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