Properties

Label 2-60840-1.1-c1-0-25
Degree $2$
Conductor $60840$
Sign $1$
Analytic cond. $485.809$
Root an. cond. $22.0410$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5-s − 4·7-s + 4·11-s − 6·17-s + 4·23-s + 25-s + 6·29-s + 8·31-s − 4·35-s + 2·37-s + 10·41-s − 4·43-s + 8·47-s + 9·49-s + 2·53-s + 4·55-s + 4·59-s + 14·61-s + 12·67-s − 8·71-s + 10·73-s − 16·77-s − 4·83-s − 6·85-s + 10·89-s + 2·97-s + 101-s + ⋯
L(s)  = 1  + 0.447·5-s − 1.51·7-s + 1.20·11-s − 1.45·17-s + 0.834·23-s + 1/5·25-s + 1.11·29-s + 1.43·31-s − 0.676·35-s + 0.328·37-s + 1.56·41-s − 0.609·43-s + 1.16·47-s + 9/7·49-s + 0.274·53-s + 0.539·55-s + 0.520·59-s + 1.79·61-s + 1.46·67-s − 0.949·71-s + 1.17·73-s − 1.82·77-s − 0.439·83-s − 0.650·85-s + 1.05·89-s + 0.203·97-s + 0.0995·101-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60840\)    =    \(2^{3} \cdot 3^{2} \cdot 5 \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(485.809\)
Root analytic conductor: \(22.0410\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 60840,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.561427384\)
\(L(\frac12)\) \(\approx\) \(2.561427384\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
13 \( 1 \)
good7 \( 1 + 4 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−14.26430931310369, −13.67217404536830, −13.29113428471685, −12.89728579683111, −12.27641424240929, −11.90287127301077, −11.16626212769778, −10.79685817886628, −9.990601394026714, −9.735647923294735, −9.207876591407692, −8.728418107226493, −8.306771960267357, −7.254485671893503, −6.825792425887078, −6.404947805414369, −6.130766824657105, −5.298621971535807, −4.556570430207797, −4.033701346043248, −3.457447526173431, −2.582482621499014, −2.388320068550518, −1.141300428400326, −0.6187591656158080, 0.6187591656158080, 1.141300428400326, 2.388320068550518, 2.582482621499014, 3.457447526173431, 4.033701346043248, 4.556570430207797, 5.298621971535807, 6.130766824657105, 6.404947805414369, 6.825792425887078, 7.254485671893503, 8.306771960267357, 8.728418107226493, 9.207876591407692, 9.735647923294735, 9.990601394026714, 10.79685817886628, 11.16626212769778, 11.90287127301077, 12.27641424240929, 12.89728579683111, 13.29113428471685, 13.67217404536830, 14.26430931310369

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