Properties

Label 2-3240-1.1-c1-0-35
Degree $2$
Conductor $3240$
Sign $-1$
Analytic cond. $25.8715$
Root an. cond. $5.08640$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5-s + 11-s − 7·19-s + 6·23-s + 25-s − 7·29-s + 31-s − 2·37-s + 9·41-s − 6·43-s − 2·47-s − 7·49-s − 55-s + 3·59-s − 10·61-s − 2·67-s + 71-s + 4·79-s − 6·83-s − 7·89-s + 7·95-s + 2·97-s − 9·101-s − 6·103-s − 2·107-s + 3·109-s − 6·115-s + ⋯
L(s)  = 1  − 0.447·5-s + 0.301·11-s − 1.60·19-s + 1.25·23-s + 1/5·25-s − 1.29·29-s + 0.179·31-s − 0.328·37-s + 1.40·41-s − 0.914·43-s − 0.291·47-s − 49-s − 0.134·55-s + 0.390·59-s − 1.28·61-s − 0.244·67-s + 0.118·71-s + 0.450·79-s − 0.658·83-s − 0.741·89-s + 0.718·95-s + 0.203·97-s − 0.895·101-s − 0.591·103-s − 0.193·107-s + 0.287·109-s − 0.559·115-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3240\)    =    \(2^{3} \cdot 3^{4} \cdot 5\)
Sign: $-1$
Analytic conductor: \(25.8715\)
Root analytic conductor: \(5.08640\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 3240,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 + T \)
good7 \( 1 + p T^{2} \)
11 \( 1 - T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 7 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 7 T + p T^{2} \)
31 \( 1 - T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 9 T + p T^{2} \)
43 \( 1 + 6 T + p T^{2} \)
47 \( 1 + 2 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 - 3 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 - T + p T^{2} \)
73 \( 1 + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 + 7 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−8.317365240674473068336982570356, −7.54227626073568434476320254549, −6.79181964211011474267484400142, −6.12399482700722823913761483281, −5.13643882139048242593994299280, −4.34897160668441502760625912840, −3.59224386547074181210339138346, −2.59389192335401374823574987580, −1.46513944669369933280069977857, 0, 1.46513944669369933280069977857, 2.59389192335401374823574987580, 3.59224386547074181210339138346, 4.34897160668441502760625912840, 5.13643882139048242593994299280, 6.12399482700722823913761483281, 6.79181964211011474267484400142, 7.54227626073568434476320254549, 8.317365240674473068336982570356

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