Properties

Label 2-6024-1.1-c1-0-84
Degree $2$
Conductor $6024$
Sign $-1$
Analytic cond. $48.1018$
Root an. cond. $6.93555$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 0.195·5-s − 0.953·7-s + 9-s + 2.78·11-s + 0.661·13-s + 0.195·15-s + 2.87·17-s − 0.297·19-s + 0.953·21-s − 2.03·23-s − 4.96·25-s − 27-s − 7.41·29-s + 0.177·31-s − 2.78·33-s + 0.186·35-s − 4.21·37-s − 0.661·39-s + 1.77·41-s − 0.688·43-s − 0.195·45-s − 0.681·47-s − 6.09·49-s − 2.87·51-s − 2.80·53-s − 0.544·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.0873·5-s − 0.360·7-s + 0.333·9-s + 0.840·11-s + 0.183·13-s + 0.0504·15-s + 0.697·17-s − 0.0683·19-s + 0.208·21-s − 0.424·23-s − 0.992·25-s − 0.192·27-s − 1.37·29-s + 0.0318·31-s − 0.485·33-s + 0.0314·35-s − 0.693·37-s − 0.105·39-s + 0.277·41-s − 0.105·43-s − 0.0291·45-s − 0.0994·47-s − 0.870·49-s − 0.402·51-s − 0.385·53-s − 0.0734·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6024\)    =    \(2^{3} \cdot 3 \cdot 251\)
Sign: $-1$
Analytic conductor: \(48.1018\)
Root analytic conductor: \(6.93555\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 6024,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
251 \( 1 + T \)
good5 \( 1 + 0.195T + 5T^{2} \)
7 \( 1 + 0.953T + 7T^{2} \)
11 \( 1 - 2.78T + 11T^{2} \)
13 \( 1 - 0.661T + 13T^{2} \)
17 \( 1 - 2.87T + 17T^{2} \)
19 \( 1 + 0.297T + 19T^{2} \)
23 \( 1 + 2.03T + 23T^{2} \)
29 \( 1 + 7.41T + 29T^{2} \)
31 \( 1 - 0.177T + 31T^{2} \)
37 \( 1 + 4.21T + 37T^{2} \)
41 \( 1 - 1.77T + 41T^{2} \)
43 \( 1 + 0.688T + 43T^{2} \)
47 \( 1 + 0.681T + 47T^{2} \)
53 \( 1 + 2.80T + 53T^{2} \)
59 \( 1 - 14.2T + 59T^{2} \)
61 \( 1 - 6.19T + 61T^{2} \)
67 \( 1 + 6.86T + 67T^{2} \)
71 \( 1 + 6.57T + 71T^{2} \)
73 \( 1 - 6.65T + 73T^{2} \)
79 \( 1 - 9.77T + 79T^{2} \)
83 \( 1 - 2.12T + 83T^{2} \)
89 \( 1 + 7.16T + 89T^{2} \)
97 \( 1 + 3.02T + 97T^{2} \)
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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

−7.62645371790020195335464543550, −6.97320568417370656034274583946, −6.22155112119295944092167158141, −5.70262168875391325291150035364, −4.91271959468508933724640777682, −3.88613446697184964877091569883, −3.52386049077399476613049953731, −2.18829184666836658828844730492, −1.25197359274811676955344990034, 0, 1.25197359274811676955344990034, 2.18829184666836658828844730492, 3.52386049077399476613049953731, 3.88613446697184964877091569883, 4.91271959468508933724640777682, 5.70262168875391325291150035364, 6.22155112119295944092167158141, 6.97320568417370656034274583946, 7.62645371790020195335464543550

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