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Properties

Label 1-364-364.363-r0-0-0
Degree $1$
Conductor $364$
Sign $1$
Analytic cond. $1.69040$
Root an. cond. $1.69040$
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  + 3-s + 5-s + 9-s + 11-s + 15-s − 17-s − 19-s − 23-s + 25-s + 27-s + 29-s − 31-s + 33-s − 37-s + 41-s − 43-s + 45-s − 47-s − 51-s + 53-s + 55-s − 57-s − 59-s − 61-s + 67-s − 69-s + 71-s + ⋯
L(s)  = 1  + 3-s + 5-s + 9-s + 11-s + 15-s − 17-s − 19-s − 23-s + 25-s + 27-s + 29-s − 31-s + 33-s − 37-s + 41-s − 43-s + 45-s − 47-s − 51-s + 53-s + 55-s − 57-s − 59-s − 61-s + 67-s − 69-s + 71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 364 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 364 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(364\)    =    \(2^{2} \cdot 7 \cdot 13\)
Sign: $1$
Analytic conductor: \(1.69040\)
Root analytic conductor: \(1.69040\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{364} (363, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 364,\ (0:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.184045669\)
\(L(\frac12)\) \(\approx\) \(2.184045669\)
\(L(1)\) \(\approx\) \(1.688686762\)
\(L(1)\) \(\approx\) \(1.688686762\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
13 \( 1 \)
good3 \( 1 + T \)
5 \( 1 + T \)
11 \( 1 + T \)
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

−24.7090616161466808503976261842, −24.25512232638237752330775741943, −22.82027308152034567093539381010, −21.71934894365045535910553607119, −21.37062777580798528885167829503, −20.1249827993771741783900139197, −19.67402609870038690197873637101, −18.510677809773057987988370547138, −17.70161377160784577661411635780, −16.75964476684963420700860640681, −15.63922999117464940235311069773, −14.62368150045163266131509848512, −14.01028044502495278828084541885, −13.185853748128165204051284770388, −12.27895964478772839745040512955, −10.83930143482933302874545252071, −9.8806834550204880557621152078, −9.05789682604669672605887051333, −8.35484410376279186902545831257, −6.93394611387093542807073649951, −6.216276440940406941694163724379, −4.71454134922154020569269288269, −3.69264220524575879901192422770, −2.37397941506344628168556101709, −1.59459713622084569339497911385, 1.59459713622084569339497911385, 2.37397941506344628168556101709, 3.69264220524575879901192422770, 4.71454134922154020569269288269, 6.216276440940406941694163724379, 6.93394611387093542807073649951, 8.35484410376279186902545831257, 9.05789682604669672605887051333, 9.8806834550204880557621152078, 10.83930143482933302874545252071, 12.27895964478772839745040512955, 13.185853748128165204051284770388, 14.01028044502495278828084541885, 14.62368150045163266131509848512, 15.63922999117464940235311069773, 16.75964476684963420700860640681, 17.70161377160784577661411635780, 18.510677809773057987988370547138, 19.67402609870038690197873637101, 20.1249827993771741783900139197, 21.37062777580798528885167829503, 21.71934894365045535910553607119, 22.82027308152034567093539381010, 24.25512232638237752330775741943, 24.7090616161466808503976261842

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