Properties

Label 2-3240-1.1-c1-0-7
Degree $2$
Conductor $3240$
Sign $1$
Analytic cond. $25.8715$
Root an. cond. $5.08640$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5-s − 2.37·7-s + 3.37·11-s + 2.37·13-s − 4.74·17-s + 19-s + 0.372·23-s + 25-s + 3.37·29-s − 6.11·31-s + 2.37·35-s + 6·37-s − 11.7·41-s + 6.74·43-s + 3.62·47-s − 1.37·49-s + 7.11·53-s − 3.37·55-s − 5·59-s + 1.25·61-s − 2.37·65-s + 10.7·67-s − 1.37·71-s + 3.25·73-s − 8·77-s + 8.74·79-s + 10·83-s + ⋯
L(s)  = 1  − 0.447·5-s − 0.896·7-s + 1.01·11-s + 0.657·13-s − 1.15·17-s + 0.229·19-s + 0.0776·23-s + 0.200·25-s + 0.626·29-s − 1.09·31-s + 0.400·35-s + 0.986·37-s − 1.83·41-s + 1.02·43-s + 0.529·47-s − 0.196·49-s + 0.977·53-s − 0.454·55-s − 0.650·59-s + 0.160·61-s − 0.294·65-s + 1.31·67-s − 0.162·71-s + 0.381·73-s − 0.911·77-s + 0.983·79-s + 1.09·83-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: no
Arithmetic: yes
Character: $\chi_{3240} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3240,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.454197669\)
\(L(\frac12)\) \(\approx\) \(1.454197669\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.37T + 7T^{2} \)
11 \( 1 - 3.37T + 11T^{2} \)
13 \( 1 - 2.37T + 13T^{2} \)
17 \( 1 + 4.74T + 17T^{2} \)
19 \( 1 - T + 19T^{2} \)
23 \( 1 - 0.372T + 23T^{2} \)
29 \( 1 - 3.37T + 29T^{2} \)
31 \( 1 + 6.11T + 31T^{2} \)
37 \( 1 - 6T + 37T^{2} \)
41 \( 1 + 11.7T + 41T^{2} \)
43 \( 1 - 6.74T + 43T^{2} \)
47 \( 1 - 3.62T + 47T^{2} \)
53 \( 1 - 7.11T + 53T^{2} \)
59 \( 1 + 5T + 59T^{2} \)
61 \( 1 - 1.25T + 61T^{2} \)
67 \( 1 - 10.7T + 67T^{2} \)
71 \( 1 + 1.37T + 71T^{2} \)
73 \( 1 - 3.25T + 73T^{2} \)
79 \( 1 - 8.74T + 79T^{2} \)
83 \( 1 - 10T + 83T^{2} \)
89 \( 1 + 1.37T + 89T^{2} \)
97 \( 1 - 6.74T + 97T^{2} \)
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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

−8.823515612491441912882040047812, −7.944951697660732035604283625654, −6.95279808740597754452538918287, −6.55361472679277518570412389505, −5.77261846961797279458684752678, −4.66227671868133182889211176051, −3.86123229693247741638981815478, −3.24813357152975580898265374995, −2.05151696499587778666880325446, −0.71987161036480443378039294940, 0.71987161036480443378039294940, 2.05151696499587778666880325446, 3.24813357152975580898265374995, 3.86123229693247741638981815478, 4.66227671868133182889211176051, 5.77261846961797279458684752678, 6.55361472679277518570412389505, 6.95279808740597754452538918287, 7.944951697660732035604283625654, 8.823515612491441912882040047812

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