Properties

Label 2-3120-1.1-c1-0-5
Degree $2$
Conductor $3120$
Sign $1$
Analytic cond. $24.9133$
Root an. cond. $4.99132$
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  − 3-s − 5-s − 2.61·7-s + 9-s + 6.39·11-s + 13-s + 15-s − 0.615·17-s − 7.77·19-s + 2.61·21-s − 2.61·23-s + 25-s − 27-s + 7.00·29-s − 5.23·31-s − 6.39·33-s + 2.61·35-s + 7.16·37-s − 39-s + 7.16·41-s − 4·43-s − 45-s + 1.23·47-s − 0.161·49-s + 0.615·51-s − 13.6·53-s − 6.39·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s − 0.988·7-s + 0.333·9-s + 1.92·11-s + 0.277·13-s + 0.258·15-s − 0.149·17-s − 1.78·19-s + 0.570·21-s − 0.545·23-s + 0.200·25-s − 0.192·27-s + 1.30·29-s − 0.939·31-s − 1.11·33-s + 0.442·35-s + 1.17·37-s − 0.160·39-s + 1.11·41-s − 0.609·43-s − 0.149·45-s + 0.179·47-s − 0.0230·49-s + 0.0861·51-s − 1.87·53-s − 0.861·55-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: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3120,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.129951639\)
\(L(\frac12)\) \(\approx\) \(1.129951639\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.61T + 7T^{2} \)
11 \( 1 - 6.39T + 11T^{2} \)
17 \( 1 + 0.615T + 17T^{2} \)
19 \( 1 + 7.77T + 19T^{2} \)
23 \( 1 + 2.61T + 23T^{2} \)
29 \( 1 - 7.00T + 29T^{2} \)
31 \( 1 + 5.23T + 31T^{2} \)
37 \( 1 - 7.16T + 37T^{2} \)
41 \( 1 - 7.16T + 41T^{2} \)
43 \( 1 + 4T + 43T^{2} \)
47 \( 1 - 1.23T + 47T^{2} \)
53 \( 1 + 13.6T + 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 + 0.391T + 61T^{2} \)
67 \( 1 - 14.0T + 67T^{2} \)
71 \( 1 + 5.16T + 71T^{2} \)
73 \( 1 - 0.993T + 73T^{2} \)
79 \( 1 - 14.3T + 79T^{2} \)
83 \( 1 + 18.0T + 83T^{2} \)
89 \( 1 - 14.8T + 89T^{2} \)
97 \( 1 - 7.38T + 97T^{2} \)
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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

−8.792720284769265885999232785468, −7.970047730464109918046349024333, −6.88001119839502555977272393857, −6.39370396633059569680846969807, −6.02591709384105128700720491627, −4.59018588183955064901716931725, −4.07202580031588541264983123107, −3.28655186559322266382989149359, −1.93083545985139733897075638198, −0.66172664170434032537120859125, 0.66172664170434032537120859125, 1.93083545985139733897075638198, 3.28655186559322266382989149359, 4.07202580031588541264983123107, 4.59018588183955064901716931725, 6.02591709384105128700720491627, 6.39370396633059569680846969807, 6.88001119839502555977272393857, 7.970047730464109918046349024333, 8.792720284769265885999232785468

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