Properties

Label 1-227-227.226-r1-0-0
Degree $1$
Conductor $227$
Sign $1$
Analytic cond. $24.3945$
Root an. cond. $24.3945$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 3-s + 4-s − 5-s − 6-s + 7-s − 8-s + 9-s + 10-s + 11-s + 12-s − 13-s − 14-s − 15-s + 16-s − 17-s − 18-s + 19-s − 20-s + 21-s − 22-s + 23-s − 24-s + 25-s + 26-s + 27-s + 28-s + ⋯
L(s)  = 1  − 2-s + 3-s + 4-s − 5-s − 6-s + 7-s − 8-s + 9-s + 10-s + 11-s + 12-s − 13-s − 14-s − 15-s + 16-s − 17-s − 18-s + 19-s − 20-s + 21-s − 22-s + 23-s − 24-s + 25-s + 26-s + 27-s + 28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 227 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 227 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(227\)
Sign: $1$
Analytic conductor: \(24.3945\)
Root analytic conductor: \(24.3945\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{227} (226, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 227,\ (1:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.746062087\)
\(L(\frac12)\) \(\approx\) \(1.746062087\)
\(L(1)\) \(\approx\) \(1.042574139\)
\(L(1)\) \(\approx\) \(1.042574139\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad227 \( 1 \)
good2 \( 1 - T \)
3 \( 1 + T \)
5 \( 1 - T \)
7 \( 1 + T \)
11 \( 1 + T \)
13 \( 1 - T \)
17 \( 1 - T \)
19 \( 1 + T \)
23 \( 1 + T \)
29 \( 1 + T \)
31 \( 1 - T \)
37 \( 1 - T \)
41 \( 1 - T \)
43 \( 1 + T \)
47 \( 1 + T \)
53 \( 1 + T \)
59 \( 1 + T \)
61 \( 1 - T \)
67 \( 1 - T \)
71 \( 1 + T \)
73 \( 1 + T \)
79 \( 1 + T \)
83 \( 1 - T \)
89 \( 1 + T \)
97 \( 1 + T \)
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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

−26.512061577034567642790675601, −25.232294821693138918001093951, −24.461901395436914834422505747211, −23.99934175009914856513859678556, −22.275693809268395778558549871019, −21.13472827219716190192739824318, −20.10052102541872956701878073610, −19.74417749994118174083404945328, −18.81857565034216215600169537340, −17.81831877851907389331928539727, −16.79069219842832826952037451896, −15.5832005101008724828688303754, −14.9620935623522753043033965611, −14.08366466071811479349226165518, −12.357722978746647108889391658857, −11.570946556842816551920995940405, −10.51975656754414363669118992126, −9.16143782949255695075342973275, −8.60355748416399363056897584061, −7.49087200252593276936746632080, −6.98485679544884034211125028637, −4.81687184042712103618980088058, −3.5390428048487967663480720486, −2.26146006995272507036991556192, −0.98380091960824979110424396224, 0.98380091960824979110424396224, 2.26146006995272507036991556192, 3.5390428048487967663480720486, 4.81687184042712103618980088058, 6.98485679544884034211125028637, 7.49087200252593276936746632080, 8.60355748416399363056897584061, 9.16143782949255695075342973275, 10.51975656754414363669118992126, 11.570946556842816551920995940405, 12.357722978746647108889391658857, 14.08366466071811479349226165518, 14.9620935623522753043033965611, 15.5832005101008724828688303754, 16.79069219842832826952037451896, 17.81831877851907389331928539727, 18.81857565034216215600169537340, 19.74417749994118174083404945328, 20.10052102541872956701878073610, 21.13472827219716190192739824318, 22.275693809268395778558549871019, 23.99934175009914856513859678556, 24.461901395436914834422505747211, 25.232294821693138918001093951, 26.512061577034567642790675601

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