Properties

Label 2-1840-1.1-c1-0-25
Degree $2$
Conductor $1840$
Sign $1$
Analytic cond. $14.6924$
Root an. cond. $3.83307$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.56·3-s + 5-s + 3.12·7-s − 0.561·9-s + 4·11-s + 3.56·13-s + 1.56·15-s + 5.12·17-s − 4·19-s + 4.87·21-s − 23-s + 25-s − 5.56·27-s − 4.43·29-s − 5.56·31-s + 6.24·33-s + 3.12·35-s + 1.12·37-s + 5.56·39-s − 3.56·41-s + 0.876·43-s − 0.561·45-s − 8.68·47-s + 2.75·49-s + 8·51-s + 12.2·53-s + 4·55-s + ⋯
L(s)  = 1  + 0.901·3-s + 0.447·5-s + 1.18·7-s − 0.187·9-s + 1.20·11-s + 0.987·13-s + 0.403·15-s + 1.24·17-s − 0.917·19-s + 1.06·21-s − 0.208·23-s + 0.200·25-s − 1.07·27-s − 0.824·29-s − 0.998·31-s + 1.08·33-s + 0.527·35-s + 0.184·37-s + 0.890·39-s − 0.556·41-s + 0.133·43-s − 0.0837·45-s − 1.26·47-s + 0.393·49-s + 1.12·51-s + 1.68·53-s + 0.539·55-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: no
Arithmetic: yes
Character: $\chi_{1840} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1840,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.160510287\)
\(L(\frac12)\) \(\approx\) \(3.160510287\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.56T + 3T^{2} \)
7 \( 1 - 3.12T + 7T^{2} \)
11 \( 1 - 4T + 11T^{2} \)
13 \( 1 - 3.56T + 13T^{2} \)
17 \( 1 - 5.12T + 17T^{2} \)
19 \( 1 + 4T + 19T^{2} \)
29 \( 1 + 4.43T + 29T^{2} \)
31 \( 1 + 5.56T + 31T^{2} \)
37 \( 1 - 1.12T + 37T^{2} \)
41 \( 1 + 3.56T + 41T^{2} \)
43 \( 1 - 0.876T + 43T^{2} \)
47 \( 1 + 8.68T + 47T^{2} \)
53 \( 1 - 12.2T + 53T^{2} \)
59 \( 1 + 10.2T + 59T^{2} \)
61 \( 1 - 2.87T + 61T^{2} \)
67 \( 1 - 10.2T + 67T^{2} \)
71 \( 1 - 8.68T + 71T^{2} \)
73 \( 1 - 12.4T + 73T^{2} \)
79 \( 1 + 6.24T + 79T^{2} \)
83 \( 1 + 12T + 83T^{2} \)
89 \( 1 - 10T + 89T^{2} \)
97 \( 1 - 0.246T + 97T^{2} \)
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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

−9.138048479014353908763125874012, −8.404648144331914071340629178515, −8.007201064673294023441232040138, −6.95431654726419035953844787449, −5.99050486145251041899816503476, −5.26582250481388890386194941941, −4.04165045002945825882640415983, −3.43430068311471163271613404454, −2.10513828226632245008918839649, −1.36950685698706190392224113593, 1.36950685698706190392224113593, 2.10513828226632245008918839649, 3.43430068311471163271613404454, 4.04165045002945825882640415983, 5.26582250481388890386194941941, 5.99050486145251041899816503476, 6.95431654726419035953844787449, 8.007201064673294023441232040138, 8.404648144331914071340629178515, 9.138048479014353908763125874012

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