Properties

Label 2-1840-1.1-c1-0-22
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  + 2.79·3-s − 5-s + 1.79·7-s + 4.79·9-s + 0.791·11-s + 5.79·13-s − 2.79·15-s + 0.791·17-s − 5.79·19-s + 5·21-s − 23-s + 25-s + 4.99·27-s + 7.58·29-s + 3.37·31-s + 2.20·33-s − 1.79·35-s − 4·37-s + 16.1·39-s − 6.79·41-s − 11.1·43-s − 4.79·45-s + 4.41·47-s − 3.79·49-s + 2.20·51-s + 6·53-s − 0.791·55-s + ⋯
L(s)  = 1  + 1.61·3-s − 0.447·5-s + 0.677·7-s + 1.59·9-s + 0.238·11-s + 1.60·13-s − 0.720·15-s + 0.191·17-s − 1.32·19-s + 1.09·21-s − 0.208·23-s + 0.200·25-s + 0.962·27-s + 1.40·29-s + 0.605·31-s + 0.384·33-s − 0.302·35-s − 0.657·37-s + 2.58·39-s − 1.06·41-s − 1.70·43-s − 0.714·45-s + 0.644·47-s − 0.541·49-s + 0.309·51-s + 0.824·53-s − 0.106·55-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

\(L(1)\) \(\approx\) \(3.331190207\)
\(L(\frac12)\) \(\approx\) \(3.331190207\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.79T + 3T^{2} \)
7 \( 1 - 1.79T + 7T^{2} \)
11 \( 1 - 0.791T + 11T^{2} \)
13 \( 1 - 5.79T + 13T^{2} \)
17 \( 1 - 0.791T + 17T^{2} \)
19 \( 1 + 5.79T + 19T^{2} \)
29 \( 1 - 7.58T + 29T^{2} \)
31 \( 1 - 3.37T + 31T^{2} \)
37 \( 1 + 4T + 37T^{2} \)
41 \( 1 + 6.79T + 41T^{2} \)
43 \( 1 + 11.1T + 43T^{2} \)
47 \( 1 - 4.41T + 47T^{2} \)
53 \( 1 - 6T + 53T^{2} \)
59 \( 1 - 13.5T + 59T^{2} \)
61 \( 1 - 10.3T + 61T^{2} \)
67 \( 1 + 11.1T + 67T^{2} \)
71 \( 1 + 8.37T + 71T^{2} \)
73 \( 1 - 12.7T + 73T^{2} \)
79 \( 1 + 8T + 79T^{2} \)
83 \( 1 - 6T + 83T^{2} \)
89 \( 1 - 15.1T + 89T^{2} \)
97 \( 1 + 7.95T + 97T^{2} \)
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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

−8.744147764448629160552381936385, −8.515253642608303670110376157663, −8.105654653243987316633960702933, −7.02393613300488824488128644459, −6.28370926587918452379634881985, −4.90763234032842049286194293711, −3.97925541850641924941011032608, −3.43266895891183257743930666625, −2.31734844278758075461923703741, −1.32419911688698329532723629697, 1.32419911688698329532723629697, 2.31734844278758075461923703741, 3.43266895891183257743930666625, 3.97925541850641924941011032608, 4.90763234032842049286194293711, 6.28370926587918452379634881985, 7.02393613300488824488128644459, 8.105654653243987316633960702933, 8.515253642608303670110376157663, 8.744147764448629160552381936385

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