Properties

Label 2-5355-1.1-c1-0-25
Degree $2$
Conductor $5355$
Sign $1$
Analytic cond. $42.7598$
Root an. cond. $6.53910$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·4-s + 5-s + 7-s − 2·11-s − 5·13-s + 4·16-s − 17-s + 2·19-s − 2·20-s + 23-s + 25-s − 2·28-s − 8·29-s + 31-s + 35-s − 3·37-s + 7·41-s + 4·44-s + 47-s + 49-s + 10·52-s + 8·53-s − 2·55-s − 7·61-s − 8·64-s − 5·65-s + 16·67-s + ⋯
L(s)  = 1  − 4-s + 0.447·5-s + 0.377·7-s − 0.603·11-s − 1.38·13-s + 16-s − 0.242·17-s + 0.458·19-s − 0.447·20-s + 0.208·23-s + 1/5·25-s − 0.377·28-s − 1.48·29-s + 0.179·31-s + 0.169·35-s − 0.493·37-s + 1.09·41-s + 0.603·44-s + 0.145·47-s + 1/7·49-s + 1.38·52-s + 1.09·53-s − 0.269·55-s − 0.896·61-s − 64-s − 0.620·65-s + 1.95·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5355 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5355 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5355\)    =    \(3^{2} \cdot 5 \cdot 7 \cdot 17\)
Sign: $1$
Analytic conductor: \(42.7598\)
Root analytic conductor: \(6.53910\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 5355,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.239238697\)
\(L(\frac12)\) \(\approx\) \(1.239238697\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 - T \)
7 \( 1 - T \)
17 \( 1 + T \)
good2 \( 1 + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
13 \( 1 + 5 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 - T + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 - T + p T^{2} \)
37 \( 1 + 3 T + p T^{2} \)
41 \( 1 - 7 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 - T + p T^{2} \)
53 \( 1 - 8 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 7 T + p T^{2} \)
67 \( 1 - 16 T + p T^{2} \)
71 \( 1 - 10 T + p T^{2} \)
73 \( 1 + 8 T + p T^{2} \)
79 \( 1 + 6 T + p T^{2} \)
83 \( 1 - 13 T + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
97 \( 1 + 6 T + p T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.106487158050962269682243926675, −7.60510118066806050686071754365, −6.87791714938848219971937454293, −5.70226556815601305222886081684, −5.27735130938913567985494651166, −4.64296374829682455447246102514, −3.81822570519668162360175033796, −2.78530579677966525992222851443, −1.91627444815744177034429417146, −0.59379948344703753005746355543, 0.59379948344703753005746355543, 1.91627444815744177034429417146, 2.78530579677966525992222851443, 3.81822570519668162360175033796, 4.64296374829682455447246102514, 5.27735130938913567985494651166, 5.70226556815601305222886081684, 6.87791714938848219971937454293, 7.60510118066806050686071754365, 8.106487158050962269682243926675

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