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Properties

Label 2-60840-1.1-c1-0-31
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 + 3·7-s + 5·11-s − 5·17-s + 19-s + 23-s + 25-s + 9·29-s + 4·31-s + 3·35-s + 3·37-s − 41-s − 3·43-s − 8·47-s + 2·49-s − 10·53-s + 5·55-s + 3·59-s + 7·61-s + 9·67-s + 7·71-s − 10·73-s + 15·77-s − 16·79-s + 12·83-s − 5·85-s + 15·89-s + ⋯
L(s)  = 1  + 0.447·5-s + 1.13·7-s + 1.50·11-s − 1.21·17-s + 0.229·19-s + 0.208·23-s + 1/5·25-s + 1.67·29-s + 0.718·31-s + 0.507·35-s + 0.493·37-s − 0.156·41-s − 0.457·43-s − 1.16·47-s + 2/7·49-s − 1.37·53-s + 0.674·55-s + 0.390·59-s + 0.896·61-s + 1.09·67-s + 0.830·71-s − 1.17·73-s + 1.70·77-s − 1.80·79-s + 1.31·83-s − 0.542·85-s + 1.58·89-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

\(L(1)\) \(\approx\) \(4.025286960\)
\(L(\frac12)\) \(\approx\) \(4.025286960\)
\(L(\frac{3}{2})\) 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 \( 1 - 3 T + p T^{2} \)
11 \( 1 - 5 T + p T^{2} \)
17 \( 1 + 5 T + p T^{2} \)
19 \( 1 - T + p T^{2} \)
23 \( 1 - T + p T^{2} \)
29 \( 1 - 9 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 3 T + p T^{2} \)
41 \( 1 + T + p T^{2} \)
43 \( 1 + 3 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 - 3 T + p T^{2} \)
61 \( 1 - 7 T + p T^{2} \)
67 \( 1 - 9 T + p T^{2} \)
71 \( 1 - 7 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 15 T + p T^{2} \)
97 \( 1 - 7 T + p T^{2} \)
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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

−14.36999452253120, −13.98245643511540, −13.23309197596011, −12.95866397710339, −12.09584456818377, −11.60643542024158, −11.48467512753715, −10.76138162693826, −10.25634656317411, −9.581695193615496, −9.213215957227980, −8.426213734580950, −8.371501112812523, −7.554915957360483, −6.709103629372481, −6.564702960710634, −5.976316166293907, −4.993341123084377, −4.771338464524764, −4.193295190854679, −3.454328768259540, −2.680122628644045, −1.949038155026593, −1.406354005859407, −0.7235255458280209, 0.7235255458280209, 1.406354005859407, 1.949038155026593, 2.680122628644045, 3.454328768259540, 4.193295190854679, 4.771338464524764, 4.993341123084377, 5.976316166293907, 6.564702960710634, 6.709103629372481, 7.554915957360483, 8.371501112812523, 8.426213734580950, 9.213215957227980, 9.581695193615496, 10.25634656317411, 10.76138162693826, 11.48467512753715, 11.60643542024158, 12.09584456818377, 12.95866397710339, 13.23309197596011, 13.98245643511540, 14.36999452253120

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