Properties

Label 2-1840-460.459-c1-0-56
Degree $2$
Conductor $1840$
Sign $i$
Analytic cond. $14.6924$
Root an. cond. $3.83307$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2.23·5-s − 4.79i·7-s − 3·9-s + 6.70·17-s + 4.79i·23-s + 5.00·25-s − 29-s − 10.7i·31-s − 10.7i·35-s − 11.1·37-s + 7·41-s − 9.59i·43-s − 6.70·45-s − 15.9·49-s − 2.23·53-s + ⋯
L(s)  = 1  + 0.999·5-s − 1.81i·7-s − 9-s + 1.62·17-s + 0.999i·23-s + 1.00·25-s − 0.185·29-s − 1.92i·31-s − 1.81i·35-s − 1.83·37-s + 1.09·41-s − 1.46i·43-s − 0.999·45-s − 2.28·49-s − 0.307·53-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1840\)    =    \(2^{4} \cdot 5 \cdot 23\)
Sign: $i$
Analytic conductor: \(14.6924\)
Root analytic conductor: \(3.83307\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1840} (1839, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1840,\ (\ :1/2),\ i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.796065889\)
\(L(\frac12)\) \(\approx\) \(1.796065889\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.23T \)
23 \( 1 - 4.79iT \)
good3 \( 1 + 3T^{2} \)
7 \( 1 + 4.79iT - 7T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 - 6.70T + 17T^{2} \)
19 \( 1 + 19T^{2} \)
29 \( 1 + T + 29T^{2} \)
31 \( 1 + 10.7iT - 31T^{2} \)
37 \( 1 + 11.1T + 37T^{2} \)
41 \( 1 - 7T + 41T^{2} \)
43 \( 1 + 9.59iT - 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + 2.23T + 53T^{2} \)
59 \( 1 + 10.7iT - 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 + 4.79iT - 67T^{2} \)
71 \( 1 + 10.7iT - 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 - 14.3iT - 83T^{2} \)
89 \( 1 - 89T^{2} \)
97 \( 1 + 4.47T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.299209450648192008064914590897, −8.102077742648954047589389147469, −7.52940697822571749094622744946, −6.68619762217446445948602408757, −5.75400720650370483794534513802, −5.17440272046197004690524038794, −3.90370225711260711889642521196, −3.20865053416013096390314620718, −1.84641117177676497775036772275, −0.66524876698138781753772870351, 1.52126339665491535471055152381, 2.66608284446246635060198402759, 3.12958522663944159721637292391, 4.90585800555690236179413115686, 5.60355237382305888174986188154, 5.93768865842394703826656149652, 6.90420301742688758940700906085, 8.208735128305660902665519157606, 8.721954599538942267541681660837, 9.294533422397190699472158881012

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