Properties

Label 2-3120-1.1-c1-0-42
Degree $2$
Conductor $3120$
Sign $-1$
Analytic cond. $24.9133$
Root an. cond. $4.99132$
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  + 3-s − 5-s + 9-s − 13-s − 15-s − 6·17-s + 4·23-s + 25-s + 27-s − 10·29-s − 6·37-s − 39-s + 2·41-s + 4·43-s − 45-s − 7·49-s − 6·51-s − 6·53-s + 6·61-s + 65-s − 4·67-s + 4·69-s − 16·71-s − 2·73-s + 75-s + 81-s − 4·83-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s + 1/3·9-s − 0.277·13-s − 0.258·15-s − 1.45·17-s + 0.834·23-s + 1/5·25-s + 0.192·27-s − 1.85·29-s − 0.986·37-s − 0.160·39-s + 0.312·41-s + 0.609·43-s − 0.149·45-s − 49-s − 0.840·51-s − 0.824·53-s + 0.768·61-s + 0.124·65-s − 0.488·67-s + 0.481·69-s − 1.89·71-s − 0.234·73-s + 0.115·75-s + 1/9·81-s − 0.439·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3120\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 13\)
Sign: $-1$
Analytic conductor: \(24.9133\)
Root analytic conductor: \(4.99132\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{3120} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 3120,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
5 \( 1 + T \)
13 \( 1 + T \)
good7 \( 1 + p T^{2} \)
11 \( 1 + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 10 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 16 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−8.416060489621200118497940918415, −7.48651432603210725392086041861, −7.05881776326155092462693764661, −6.13130217685388014086757536825, −5.09225276175166925177215351932, −4.33322549174996816277799124115, −3.53361530480953533803089286496, −2.61347722092406101275253466214, −1.63593081225043651097784490840, 0, 1.63593081225043651097784490840, 2.61347722092406101275253466214, 3.53361530480953533803089286496, 4.33322549174996816277799124115, 5.09225276175166925177215351932, 6.13130217685388014086757536825, 7.05881776326155092462693764661, 7.48651432603210725392086041861, 8.416060489621200118497940918415

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