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Properties

Label 2-59840-1.1-c1-0-3
Degree $2$
Conductor $59840$
Sign $1$
Analytic cond. $477.824$
Root an. cond. $21.8592$
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 − 3·9-s + 11-s − 2·13-s − 17-s − 8·23-s + 25-s + 10·29-s − 6·37-s + 6·41-s − 4·43-s + 3·45-s + 12·47-s − 7·49-s + 6·53-s − 55-s − 12·59-s − 14·61-s + 2·65-s − 4·67-s + 8·71-s − 10·73-s + 4·79-s + 9·81-s − 12·83-s + 85-s + 10·89-s + ⋯
L(s)  = 1  − 0.447·5-s − 9-s + 0.301·11-s − 0.554·13-s − 0.242·17-s − 1.66·23-s + 1/5·25-s + 1.85·29-s − 0.986·37-s + 0.937·41-s − 0.609·43-s + 0.447·45-s + 1.75·47-s − 49-s + 0.824·53-s − 0.134·55-s − 1.56·59-s − 1.79·61-s + 0.248·65-s − 0.488·67-s + 0.949·71-s − 1.17·73-s + 0.450·79-s + 81-s − 1.31·83-s + 0.108·85-s + 1.05·89-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 59840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 59840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(59840\)    =    \(2^{6} \cdot 5 \cdot 11 \cdot 17\)
Sign: $1$
Analytic conductor: \(477.824\)
Root analytic conductor: \(21.8592\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 59840,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7780981225\)
\(L(\frac12)\) \(\approx\) \(0.7780981225\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + T \)
11 \( 1 - T \)
17 \( 1 + T \)
good3 \( 1 + p T^{2} \)
7 \( 1 + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 - 10 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 + 18 T + p T^{2} \)
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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

−14.08159895904293, −13.98647170217153, −13.54822280640295, −12.48719707476962, −12.24953319414486, −11.92981574391498, −11.33101978759538, −10.70660582844265, −10.35412108809585, −9.657083023612529, −9.158389797716906, −8.450348445310781, −8.279680218115031, −7.530386226628240, −7.079849693792706, −6.227335166870313, −6.034603764373451, −5.247137270677696, −4.575744205518090, −4.162117540219440, −3.354694660490401, −2.804869380434337, −2.192386987549236, −1.313746087416468, −0.3045627779647973, 0.3045627779647973, 1.313746087416468, 2.192386987549236, 2.804869380434337, 3.354694660490401, 4.162117540219440, 4.575744205518090, 5.247137270677696, 6.034603764373451, 6.227335166870313, 7.079849693792706, 7.530386226628240, 8.279680218115031, 8.450348445310781, 9.158389797716906, 9.657083023612529, 10.35412108809585, 10.70660582844265, 11.33101978759538, 11.92981574391498, 12.24953319414486, 12.48719707476962, 13.54822280640295, 13.98647170217153, 14.08159895904293

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