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Properties

Label 2-60984-1.1-c1-0-6
Degree $2$
Conductor $60984$
Sign $1$
Analytic cond. $486.959$
Root an. cond. $22.0671$
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  + 3·5-s − 7-s − 3·13-s − 4·17-s + 19-s − 4·23-s + 4·25-s + 3·29-s − 2·31-s − 3·35-s − 3·37-s + 4·43-s − 3·47-s + 49-s + 10·53-s − 3·59-s − 2·61-s − 9·65-s + 9·67-s + 16·71-s − 5·73-s − 6·79-s − 16·83-s − 12·85-s + 6·89-s + 3·91-s + 3·95-s + ⋯
L(s)  = 1  + 1.34·5-s − 0.377·7-s − 0.832·13-s − 0.970·17-s + 0.229·19-s − 0.834·23-s + 4/5·25-s + 0.557·29-s − 0.359·31-s − 0.507·35-s − 0.493·37-s + 0.609·43-s − 0.437·47-s + 1/7·49-s + 1.37·53-s − 0.390·59-s − 0.256·61-s − 1.11·65-s + 1.09·67-s + 1.89·71-s − 0.585·73-s − 0.675·79-s − 1.75·83-s − 1.30·85-s + 0.635·89-s + 0.314·91-s + 0.307·95-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60984 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60984 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60984\)    =    \(2^{3} \cdot 3^{2} \cdot 7 \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(486.959\)
Root analytic conductor: \(22.0671\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 60984,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.157668144\)
\(L(\frac12)\) \(\approx\) \(2.157668144\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 + T \)
11 \( 1 \)
good5 \( 1 - 3 T + p T^{2} \)
13 \( 1 + 3 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 - T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 + 3 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 3 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 + 3 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 9 T + p T^{2} \)
71 \( 1 - 16 T + p T^{2} \)
73 \( 1 + 5 T + p T^{2} \)
79 \( 1 + 6 T + p T^{2} \)
83 \( 1 + 16 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 4 T + p T^{2} \)
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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

−14.11612313675575, −13.92124730543884, −13.18685368637077, −12.99791810418141, −12.26654222014124, −11.91279424653743, −11.16554748394230, −10.61218998864289, −10.14686537573517, −9.591141921606016, −9.423692709995644, −8.642917161313170, −8.234228472913517, −7.334870720196809, −6.949894323399946, −6.357242971315282, −5.846416514993633, −5.358326138255928, −4.739457795785296, −4.113495280145148, −3.350540742187911, −2.509891997009160, −2.215193218396677, −1.489260632526651, −0.4784372365050363, 0.4784372365050363, 1.489260632526651, 2.215193218396677, 2.509891997009160, 3.350540742187911, 4.113495280145148, 4.739457795785296, 5.358326138255928, 5.846416514993633, 6.357242971315282, 6.949894323399946, 7.334870720196809, 8.234228472913517, 8.642917161313170, 9.423692709995644, 9.591141921606016, 10.14686537573517, 10.61218998864289, 11.16554748394230, 11.91279424653743, 12.26654222014124, 12.99791810418141, 13.18685368637077, 13.92124730543884, 14.11612313675575

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