Properties

Label 2-1840-1.1-c1-0-42
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.14·3-s − 5-s + 1.14·7-s + 1.60·9-s − 5.89·11-s − 4.89·13-s − 2.14·15-s − 5.89·17-s + 2.34·19-s + 2.45·21-s − 23-s + 25-s − 3.00·27-s + 3.74·29-s − 5.68·31-s − 12.6·33-s − 1.14·35-s + 4·37-s − 10.4·39-s − 1.05·41-s − 11.4·43-s − 1.60·45-s − 7.74·47-s − 5.68·49-s − 12.6·51-s + 12.9·53-s + 5.89·55-s + ⋯
L(s)  = 1  + 1.23·3-s − 0.447·5-s + 0.432·7-s + 0.533·9-s − 1.77·11-s − 1.35·13-s − 0.553·15-s − 1.42·17-s + 0.538·19-s + 0.536·21-s − 0.208·23-s + 0.200·25-s − 0.577·27-s + 0.695·29-s − 1.02·31-s − 2.20·33-s − 0.193·35-s + 0.657·37-s − 1.68·39-s − 0.165·41-s − 1.75·43-s − 0.238·45-s − 1.12·47-s − 0.812·49-s − 1.76·51-s + 1.78·53-s + 0.794·55-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: \(1\)
Selberg data: \((2,\ 1840,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.14T + 3T^{2} \)
7 \( 1 - 1.14T + 7T^{2} \)
11 \( 1 + 5.89T + 11T^{2} \)
13 \( 1 + 4.89T + 13T^{2} \)
17 \( 1 + 5.89T + 17T^{2} \)
19 \( 1 - 2.34T + 19T^{2} \)
29 \( 1 - 3.74T + 29T^{2} \)
31 \( 1 + 5.68T + 31T^{2} \)
37 \( 1 - 4T + 37T^{2} \)
41 \( 1 + 1.05T + 41T^{2} \)
43 \( 1 + 11.4T + 43T^{2} \)
47 \( 1 + 7.74T + 47T^{2} \)
53 \( 1 - 12.9T + 53T^{2} \)
59 \( 1 + 0.797T + 59T^{2} \)
61 \( 1 - 13.8T + 61T^{2} \)
67 \( 1 - 15.5T + 67T^{2} \)
71 \( 1 + 2.94T + 71T^{2} \)
73 \( 1 - 6.32T + 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 - 0.912T + 83T^{2} \)
89 \( 1 + 15.4T + 89T^{2} \)
97 \( 1 - 13.3T + 97T^{2} \)
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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

−8.584447899011735667339580925616, −8.172579301285071387617011357481, −7.51702056163198319132966030292, −6.81313382867175619352484404742, −5.29990444401821595532093900436, −4.78767073186704803329905978589, −3.65522218727165515436571081491, −2.64700778070461331394599302020, −2.14756264929705504930733200813, 0, 2.14756264929705504930733200813, 2.64700778070461331394599302020, 3.65522218727165515436571081491, 4.78767073186704803329905978589, 5.29990444401821595532093900436, 6.81313382867175619352484404742, 7.51702056163198319132966030292, 8.172579301285071387617011357481, 8.584447899011735667339580925616

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