Properties

Label 2-3120-1.1-c1-0-7
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 − 3.43·7-s + 9-s + 2.63·11-s − 13-s − 15-s + 7.84·17-s − 0.794·19-s + 3.43·21-s + 3.43·23-s + 25-s − 27-s − 4.06·29-s − 9.27·31-s − 2.63·33-s − 3.43·35-s − 0.636·37-s + 39-s − 4.63·41-s − 2.41·43-s + 45-s − 5.27·47-s + 4.77·49-s − 7.84·51-s + 11.4·53-s + 2.63·55-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.447·5-s − 1.29·7-s + 0.333·9-s + 0.795·11-s − 0.277·13-s − 0.258·15-s + 1.90·17-s − 0.182·19-s + 0.748·21-s + 0.715·23-s + 0.200·25-s − 0.192·27-s − 0.755·29-s − 1.66·31-s − 0.458·33-s − 0.580·35-s − 0.104·37-s + 0.160·39-s − 0.724·41-s − 0.367·43-s + 0.149·45-s − 0.769·47-s + 0.682·49-s − 1.09·51-s + 1.57·53-s + 0.355·55-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

\(L(1)\) \(\approx\) \(1.394590876\)
\(L(\frac12)\) \(\approx\) \(1.394590876\)
\(L(\frac{3}{2})\) 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 \( 1 + 3.43T + 7T^{2} \)
11 \( 1 - 2.63T + 11T^{2} \)
17 \( 1 - 7.84T + 17T^{2} \)
19 \( 1 + 0.794T + 19T^{2} \)
23 \( 1 - 3.43T + 23T^{2} \)
29 \( 1 + 4.06T + 29T^{2} \)
31 \( 1 + 9.27T + 31T^{2} \)
37 \( 1 + 0.636T + 37T^{2} \)
41 \( 1 + 4.63T + 41T^{2} \)
43 \( 1 + 2.41T + 43T^{2} \)
47 \( 1 + 5.27T + 47T^{2} \)
53 \( 1 - 11.4T + 53T^{2} \)
59 \( 1 - 12.1T + 59T^{2} \)
61 \( 1 + 11.4T + 61T^{2} \)
67 \( 1 - 6.41T + 67T^{2} \)
71 \( 1 - 10.6T + 71T^{2} \)
73 \( 1 - 16.0T + 73T^{2} \)
79 \( 1 + 10.6T + 79T^{2} \)
83 \( 1 - 16.1T + 83T^{2} \)
89 \( 1 - 8.63T + 89T^{2} \)
97 \( 1 - 12.2T + 97T^{2} \)
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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

−8.952233691632257807270625995228, −7.79017772346827774600925464413, −6.98773614581844256125543911655, −6.46446905359685996688498538229, −5.62727672827086514213991817948, −5.13098329823999037601331615606, −3.73880879384199806928652712888, −3.31347512518719645037624715986, −1.94893545933937241479782509897, −0.73970101633406779976210526847, 0.73970101633406779976210526847, 1.94893545933937241479782509897, 3.31347512518719645037624715986, 3.73880879384199806928652712888, 5.13098329823999037601331615606, 5.62727672827086514213991817948, 6.46446905359685996688498538229, 6.98773614581844256125543911655, 7.79017772346827774600925464413, 8.952233691632257807270625995228

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