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Properties

Label 1-15e2-225.214-r0-0-0
Degree $1$
Conductor $225$
Sign $0.988 - 0.152i$
Analytic cond. $1.04489$
Root an. cond. $1.04489$
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  + (0.978 − 0.207i)2-s + (0.913 − 0.406i)4-s + (0.5 + 0.866i)7-s + (0.809 − 0.587i)8-s + (−0.978 + 0.207i)11-s + (0.978 + 0.207i)13-s + (0.669 + 0.743i)14-s + (0.669 − 0.743i)16-s + (0.809 − 0.587i)17-s + (−0.809 + 0.587i)19-s + (−0.913 + 0.406i)22-s + (−0.669 − 0.743i)23-s + 26-s + (0.809 + 0.587i)28-s + (−0.104 + 0.994i)29-s + ⋯
L(s)  = 1  + (0.978 − 0.207i)2-s + (0.913 − 0.406i)4-s + (0.5 + 0.866i)7-s + (0.809 − 0.587i)8-s + (−0.978 + 0.207i)11-s + (0.978 + 0.207i)13-s + (0.669 + 0.743i)14-s + (0.669 − 0.743i)16-s + (0.809 − 0.587i)17-s + (−0.809 + 0.587i)19-s + (−0.913 + 0.406i)22-s + (−0.669 − 0.743i)23-s + 26-s + (0.809 + 0.587i)28-s + (−0.104 + 0.994i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.988 - 0.152i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.988 - 0.152i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $0.988 - 0.152i$
Analytic conductor: \(1.04489\)
Root analytic conductor: \(1.04489\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (214, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 225,\ (0:\ ),\ 0.988 - 0.152i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.227527852 - 0.1713989270i\)
\(L(\frac12)\) \(\approx\) \(2.227527852 - 0.1713989270i\)
\(L(1)\) \(\approx\) \(1.874091157 - 0.1358618973i\)
\(L(1)\) \(\approx\) \(1.874091157 - 0.1358618973i\)

Euler product

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

−26.03897136491226037676774981853, −25.65675817803862909916279804759, −24.2492633125115298564649317384, −23.56139389357684104248483199784, −23.07717484072513837509946315365, −21.68988083388098971046020077551, −21.01041976269108601113936418154, −20.263583963457667237479463640655, −19.11387073370537874553655678277, −17.746282765460630001339735655965, −16.83978269873922642369832256104, −15.815820188377794284769185631587, −15.004197326622375305625847201638, −13.78250176570877056037990826856, −13.32704674119186308782644321170, −12.123811775933070453420041101164, −10.957634917220210290295413038457, −10.33679758587847719503699067862, −8.34153949972900650431587744968, −7.60723950620741511400228386946, −6.35586832013527445325367600414, −5.31379701422909137367773132600, −4.18770361673608115276993489147, −3.16929563878063707523207682591, −1.61684382646092698714214547560, 1.74388715969531713783670614769, 2.8222814135519528702838161386, 4.15298540875972129641490526183, 5.3332202530960785454980289480, 6.080708125218848922014695713846, 7.50138729349211524432497934883, 8.61979869826562662651147104064, 10.15030146049059083142177274704, 11.07324816308424009570388766548, 12.09830739491347494610276023828, 12.868846825251335425413914896081, 13.99501127918951590484447024144, 14.86119822010646334153083864231, 15.75739008251360478775270166406, 16.599594759266644081448437521845, 18.29230436684800647833103169359, 18.779728900084754056502381246052, 20.25614759196960817180569182211, 21.00047439165012998101478322514, 21.611803049339896364909814698215, 22.7926375180368609796511167271, 23.50410704960000417484025860460, 24.40453100217360394505603728886, 25.338787686401911125683940308656, 26.083915862449681195507984437231

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