Properties

Label 2-1840-1.1-c1-0-13
Degree $2$
Conductor $1840$
Sign $1$
Analytic cond. $14.6924$
Root an. cond. $3.83307$
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 − 7-s − 3·9-s + 6·11-s − 2·13-s − 3·17-s + 6·19-s − 23-s + 25-s + 3·29-s + 3·31-s − 35-s + 37-s + 9·41-s + 8·43-s − 3·45-s − 4·47-s − 6·49-s + 53-s + 6·55-s − 59-s + 8·61-s + 3·63-s − 2·65-s + 7·67-s + 5·71-s − 6·73-s + ⋯
L(s)  = 1  + 0.447·5-s − 0.377·7-s − 9-s + 1.80·11-s − 0.554·13-s − 0.727·17-s + 1.37·19-s − 0.208·23-s + 1/5·25-s + 0.557·29-s + 0.538·31-s − 0.169·35-s + 0.164·37-s + 1.40·41-s + 1.21·43-s − 0.447·45-s − 0.583·47-s − 6/7·49-s + 0.137·53-s + 0.809·55-s − 0.130·59-s + 1.02·61-s + 0.377·63-s − 0.248·65-s + 0.855·67-s + 0.593·71-s − 0.702·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1840\)    =    \(2^{4} \cdot 5 \cdot 23\)
Sign: $1$
Analytic conductor: \(14.6924\)
Root analytic conductor: \(3.83307\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1840} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1840,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.826020854\)
\(L(\frac12)\) \(\approx\) \(1.826020854\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
23 \( 1 + T \)
good3 \( 1 + p T^{2} \)
7 \( 1 + T + p T^{2} \)
11 \( 1 - 6 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 - 3 T + p T^{2} \)
37 \( 1 - T + p T^{2} \)
41 \( 1 - 9 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 - T + p T^{2} \)
59 \( 1 + T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 - 7 T + p T^{2} \)
71 \( 1 - 5 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 11 T + p T^{2} \)
89 \( 1 - 4 T + p T^{2} \)
97 \( 1 - 6 T + p T^{2} \)
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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

−9.370845902172061297584224195948, −8.648963889549550997042854812696, −7.66422277773345536118064776340, −6.67290253533681347908702724491, −6.18709233320869651885027817519, −5.28023148829339091268675270237, −4.26366539891827765942254240707, −3.27734500100801096074623940175, −2.32566640319124508162435295458, −0.943553537319372488144220598724, 0.943553537319372488144220598724, 2.32566640319124508162435295458, 3.27734500100801096074623940175, 4.26366539891827765942254240707, 5.28023148829339091268675270237, 6.18709233320869651885027817519, 6.67290253533681347908702724491, 7.66422277773345536118064776340, 8.648963889549550997042854812696, 9.370845902172061297584224195948

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