Properties

Label 2-3240-1.1-c1-0-11
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 + 3.37·7-s − 2.37·11-s − 3.37·13-s + 6.74·17-s + 19-s − 5.37·23-s + 25-s − 2.37·29-s + 11.1·31-s − 3.37·35-s + 6·37-s − 0.255·41-s − 4.74·43-s + 9.37·47-s + 4.37·49-s − 10.1·53-s + 2.37·55-s − 5·59-s + 12.7·61-s + 3.37·65-s − 0.744·67-s + 4.37·71-s + 14.7·73-s − 8·77-s − 2.74·79-s + 10·83-s + ⋯
L(s)  = 1  − 0.447·5-s + 1.27·7-s − 0.715·11-s − 0.935·13-s + 1.63·17-s + 0.229·19-s − 1.12·23-s + 0.200·25-s − 0.440·29-s + 1.99·31-s − 0.570·35-s + 0.986·37-s − 0.0398·41-s − 0.723·43-s + 1.36·47-s + 0.624·49-s − 1.38·53-s + 0.319·55-s − 0.650·59-s + 1.63·61-s + 0.418·65-s − 0.0909·67-s + 0.518·71-s + 1.72·73-s − 0.911·77-s − 0.308·79-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

\(L(1)\) \(\approx\) \(1.875237316\)
\(L(\frac12)\) \(\approx\) \(1.875237316\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.37T + 7T^{2} \)
11 \( 1 + 2.37T + 11T^{2} \)
13 \( 1 + 3.37T + 13T^{2} \)
17 \( 1 - 6.74T + 17T^{2} \)
19 \( 1 - T + 19T^{2} \)
23 \( 1 + 5.37T + 23T^{2} \)
29 \( 1 + 2.37T + 29T^{2} \)
31 \( 1 - 11.1T + 31T^{2} \)
37 \( 1 - 6T + 37T^{2} \)
41 \( 1 + 0.255T + 41T^{2} \)
43 \( 1 + 4.74T + 43T^{2} \)
47 \( 1 - 9.37T + 47T^{2} \)
53 \( 1 + 10.1T + 53T^{2} \)
59 \( 1 + 5T + 59T^{2} \)
61 \( 1 - 12.7T + 61T^{2} \)
67 \( 1 + 0.744T + 67T^{2} \)
71 \( 1 - 4.37T + 71T^{2} \)
73 \( 1 - 14.7T + 73T^{2} \)
79 \( 1 + 2.74T + 79T^{2} \)
83 \( 1 - 10T + 83T^{2} \)
89 \( 1 - 4.37T + 89T^{2} \)
97 \( 1 + 4.74T + 97T^{2} \)
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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

−8.282241270321047635030709090503, −7.904577922833787535071223303594, −7.53381858260770277969111023531, −6.38648965430998209541156520096, −5.39933642400471517089722842931, −4.89916663875080928341708038177, −4.07192072118069796420717186976, −2.99568365351038639136920216979, −2.06133784547470349776681212413, −0.836615048177461162379251488877, 0.836615048177461162379251488877, 2.06133784547470349776681212413, 2.99568365351038639136920216979, 4.07192072118069796420717186976, 4.89916663875080928341708038177, 5.39933642400471517089722842931, 6.38648965430998209541156520096, 7.53381858260770277969111023531, 7.904577922833787535071223303594, 8.282241270321047635030709090503

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