Properties

Label 1-1145-1145.1144-r0-0-0
Degree $1$
Conductor $1145$
Sign $1$
Analytic cond. $5.31735$
Root an. cond. $5.31735$
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 − 6-s + 7-s + 8-s + 9-s + 11-s − 12-s + 13-s + 14-s + 16-s − 17-s + 18-s + 19-s − 21-s + 22-s + 23-s − 24-s + 26-s − 27-s + 28-s − 29-s − 31-s + 32-s − 33-s − 34-s + ⋯
L(s)  = 1  + 2-s − 3-s + 4-s − 6-s + 7-s + 8-s + 9-s + 11-s − 12-s + 13-s + 14-s + 16-s − 17-s + 18-s + 19-s − 21-s + 22-s + 23-s − 24-s + 26-s − 27-s + 28-s − 29-s − 31-s + 32-s − 33-s − 34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1145 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1145 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1145\)    =    \(5 \cdot 229\)
Sign: $1$
Analytic conductor: \(5.31735\)
Root analytic conductor: \(5.31735\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1145} (1144, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 1145,\ (0:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.861491889\)
\(L(\frac12)\) \(\approx\) \(2.861491889\)
\(L(1)\) \(\approx\) \(1.850151876\)
\(L(1)\) \(\approx\) \(1.850151876\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
229 \( 1 \)
good2 \( 1 + T \)
3 \( 1 - T \)
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

−21.594021540832812203608300682557, −20.60819841966646448304383607121, −20.18518788508950556052181767277, −18.90045823468060961155687190693, −18.15287471136325937467683938092, −17.19925551140539515945064484807, −16.73128493115027668697275427837, −15.69022861632903257289560347048, −15.18840772707070224274893247479, −14.17956112430566345006411740349, −13.49194182866692446159760467939, −12.638326117655462323508973723230, −11.76886804943861869444144496396, −11.168362286202846729801371291206, −10.84269812350252494026383413922, −9.50985933794485593006639543379, −8.40014973792814400852067859395, −7.1731990858836848329356889167, −6.717369843497139389909975801103, −5.65218444699319510114451448600, −5.10133533456546418237773331502, −4.19199431334497285465653490652, −3.465716185875986126005266238769, −1.85114040266492740490138226870, −1.245047312701778168427478892934, 1.245047312701778168427478892934, 1.85114040266492740490138226870, 3.465716185875986126005266238769, 4.19199431334497285465653490652, 5.10133533456546418237773331502, 5.65218444699319510114451448600, 6.717369843497139389909975801103, 7.1731990858836848329356889167, 8.40014973792814400852067859395, 9.50985933794485593006639543379, 10.84269812350252494026383413922, 11.168362286202846729801371291206, 11.76886804943861869444144496396, 12.638326117655462323508973723230, 13.49194182866692446159760467939, 14.17956112430566345006411740349, 15.18840772707070224274893247479, 15.69022861632903257289560347048, 16.73128493115027668697275427837, 17.19925551140539515945064484807, 18.15287471136325937467683938092, 18.90045823468060961155687190693, 20.18518788508950556052181767277, 20.60819841966646448304383607121, 21.594021540832812203608300682557

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