Properties

Label 2-1840-1.1-c1-0-15
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  + 1.31·3-s − 5-s + 4.66·7-s − 1.28·9-s − 2.23·11-s − 2.80·13-s − 1.31·15-s + 7.63·17-s + 1.36·19-s + 6.11·21-s + 23-s + 25-s − 5.61·27-s + 8.94·29-s + 1.58·31-s − 2.92·33-s − 4.66·35-s − 1.40·37-s − 3.67·39-s + 10.7·41-s + 1.28·45-s + 7.26·47-s + 14.7·49-s + 10.0·51-s − 8.38·53-s + 2.23·55-s + 1.78·57-s + ⋯
L(s)  = 1  + 0.756·3-s − 0.447·5-s + 1.76·7-s − 0.427·9-s − 0.672·11-s − 0.776·13-s − 0.338·15-s + 1.85·17-s + 0.312·19-s + 1.33·21-s + 0.208·23-s + 0.200·25-s − 1.08·27-s + 1.66·29-s + 0.284·31-s − 0.508·33-s − 0.788·35-s − 0.230·37-s − 0.587·39-s + 1.67·41-s + 0.191·45-s + 1.05·47-s + 2.10·49-s + 1.40·51-s − 1.15·53-s + 0.300·55-s + 0.236·57-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

\(L(1)\) \(\approx\) \(2.450853463\)
\(L(\frac12)\) \(\approx\) \(2.450853463\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.31T + 3T^{2} \)
7 \( 1 - 4.66T + 7T^{2} \)
11 \( 1 + 2.23T + 11T^{2} \)
13 \( 1 + 2.80T + 13T^{2} \)
17 \( 1 - 7.63T + 17T^{2} \)
19 \( 1 - 1.36T + 19T^{2} \)
29 \( 1 - 8.94T + 29T^{2} \)
31 \( 1 - 1.58T + 31T^{2} \)
37 \( 1 + 1.40T + 37T^{2} \)
41 \( 1 - 10.7T + 41T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 - 7.26T + 47T^{2} \)
53 \( 1 + 8.38T + 53T^{2} \)
59 \( 1 - 4.88T + 59T^{2} \)
61 \( 1 - 4.33T + 61T^{2} \)
67 \( 1 + 8.54T + 67T^{2} \)
71 \( 1 + 8.81T + 71T^{2} \)
73 \( 1 + 5.26T + 73T^{2} \)
79 \( 1 + 7.08T + 79T^{2} \)
83 \( 1 - 4.59T + 83T^{2} \)
89 \( 1 + 4.70T + 89T^{2} \)
97 \( 1 - 16.5T + 97T^{2} \)
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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

−9.016582168823071839985359932099, −8.311455487006037228373533695297, −7.72915346197858723845644383251, −7.42892889986029071243694209985, −5.82520055560519167698584789344, −5.09825589430655363909642751251, −4.37295117066608871567554175610, −3.15386255094323939197696483215, −2.40917653730055886239035235130, −1.09845870033423592219487773786, 1.09845870033423592219487773786, 2.40917653730055886239035235130, 3.15386255094323939197696483215, 4.37295117066608871567554175610, 5.09825589430655363909642751251, 5.82520055560519167698584789344, 7.42892889986029071243694209985, 7.72915346197858723845644383251, 8.311455487006037228373533695297, 9.016582168823071839985359932099

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