Properties

Label 2-1840-23.22-c2-0-5
Degree $2$
Conductor $1840$
Sign $-0.541 - 0.840i$
Analytic cond. $50.1363$
Root an. cond. $7.08070$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 4.84·3-s − 2.23i·5-s + 4.25i·7-s + 14.4·9-s − 7.80i·11-s − 9.89·13-s + 10.8i·15-s − 23.5i·17-s + 22.1i·19-s − 20.6i·21-s + (−19.3 + 12.4i)23-s − 5.00·25-s − 26.4·27-s + 57.8·29-s − 38.0·31-s + ⋯
L(s)  = 1  − 1.61·3-s − 0.447i·5-s + 0.608i·7-s + 1.60·9-s − 0.709i·11-s − 0.760·13-s + 0.721i·15-s − 1.38i·17-s + 1.16i·19-s − 0.981i·21-s + (−0.840 + 0.541i)23-s − 0.200·25-s − 0.978·27-s + 1.99·29-s − 1.22·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.541 - 0.840i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.541 - 0.840i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1840\)    =    \(2^{4} \cdot 5 \cdot 23\)
Sign: $-0.541 - 0.840i$
Analytic conductor: \(50.1363\)
Root analytic conductor: \(7.08070\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1840} (321, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1840,\ (\ :1),\ -0.541 - 0.840i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.2026297823\)
\(L(\frac12)\) \(\approx\) \(0.2026297823\)
\(L(2)\) 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 \( 1 + 2.23iT \)
23 \( 1 + (19.3 - 12.4i)T \)
good3 \( 1 + 4.84T + 9T^{2} \)
7 \( 1 - 4.25iT - 49T^{2} \)
11 \( 1 + 7.80iT - 121T^{2} \)
13 \( 1 + 9.89T + 169T^{2} \)
17 \( 1 + 23.5iT - 289T^{2} \)
19 \( 1 - 22.1iT - 361T^{2} \)
29 \( 1 - 57.8T + 841T^{2} \)
31 \( 1 + 38.0T + 961T^{2} \)
37 \( 1 + 14.1iT - 1.36e3T^{2} \)
41 \( 1 + 18.7T + 1.68e3T^{2} \)
43 \( 1 + 56.4iT - 1.84e3T^{2} \)
47 \( 1 - 62.7T + 2.20e3T^{2} \)
53 \( 1 + 90.9iT - 2.80e3T^{2} \)
59 \( 1 + 56.2T + 3.48e3T^{2} \)
61 \( 1 - 4.12iT - 3.72e3T^{2} \)
67 \( 1 + 116. iT - 4.48e3T^{2} \)
71 \( 1 - 84.6T + 5.04e3T^{2} \)
73 \( 1 + 105.T + 5.32e3T^{2} \)
79 \( 1 - 72.7iT - 6.24e3T^{2} \)
83 \( 1 + 29.1iT - 6.88e3T^{2} \)
89 \( 1 - 106. iT - 7.92e3T^{2} \)
97 \( 1 - 94.8iT - 9.40e3T^{2} \)
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

−9.479516508926442391076352068713, −8.587344263898950952651528771327, −7.62883731719906846649509522280, −6.78867150799152788077907073135, −5.91240885261659117173838027447, −5.37997876680447852215270102371, −4.79428437501558961556456035409, −3.64303119949036034914409829421, −2.22392579567197807274811584677, −0.867078062386832006242020850461, 0.090477462273616630490924198344, 1.33787825903143890947688393565, 2.67836766209793340476617304997, 4.23235980722431256864754044911, 4.61805252422488116384676911934, 5.68027145801344289145265619900, 6.41058201994548477272368846049, 7.01385319120145901613957844659, 7.69764778491912884943553210535, 8.868055227358308369129605735579

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