Properties

Label 2-59840-1.1-c1-0-14
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 + 6·13-s + 17-s − 2·19-s − 2·23-s + 25-s − 2·29-s + 4·31-s + 2·37-s + 12·41-s − 8·43-s − 3·45-s + 8·47-s − 7·49-s − 4·53-s + 55-s − 12·59-s + 10·61-s + 6·65-s − 2·67-s − 8·71-s + 2·73-s + 10·79-s + 9·81-s + 16·83-s + ⋯
L(s)  = 1  + 0.447·5-s − 9-s + 0.301·11-s + 1.66·13-s + 0.242·17-s − 0.458·19-s − 0.417·23-s + 1/5·25-s − 0.371·29-s + 0.718·31-s + 0.328·37-s + 1.87·41-s − 1.21·43-s − 0.447·45-s + 1.16·47-s − 49-s − 0.549·53-s + 0.134·55-s − 1.56·59-s + 1.28·61-s + 0.744·65-s − 0.244·67-s − 0.949·71-s + 0.234·73-s + 1.12·79-s + 81-s + 1.75·83-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

\(L(1)\) \(\approx\) \(2.648830597\)
\(L(\frac12)\) \(\approx\) \(2.648830597\)
\(L(\frac{3}{2})\) 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 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 + 2 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 12 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 4 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 - 16 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + p T^{2} \)
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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

−14.11139044325589, −13.92765338948451, −13.36601889429006, −12.89649023460876, −12.27173697028259, −11.73379302254203, −11.19117750211260, −10.84094609159292, −10.33556819770385, −9.602527442726900, −9.078557732938159, −8.761493848085341, −7.987063014983376, −7.847308879067652, −6.764643990921798, −6.256482594734576, −6.002003765437194, −5.405639516215179, −4.674044257404618, −3.977576091681569, −3.433017710264939, −2.796773250608127, −2.076277021878212, −1.335153404835019, −0.5800494440579943, 0.5800494440579943, 1.335153404835019, 2.076277021878212, 2.796773250608127, 3.433017710264939, 3.977576091681569, 4.674044257404618, 5.405639516215179, 6.002003765437194, 6.256482594734576, 6.764643990921798, 7.847308879067652, 7.987063014983376, 8.761493848085341, 9.078557732938159, 9.602527442726900, 10.33556819770385, 10.84094609159292, 11.19117750211260, 11.73379302254203, 12.27173697028259, 12.89649023460876, 13.36601889429006, 13.92765338948451, 14.11139044325589

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