↑

Properties

Label 2-10-5.4-c11-0-4
Degree $2$
Conductor $10$
Sign $-0.803 + 0.594i$
Analytic cond. $7.68343$
Root an. cond. $2.77190$
Motivic weight $11$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 32i·2-s + 697. i·3-s − 1.02e3·4-s + (−4.15e3 − 5.61e3i)5-s + 2.23e4·6-s − 7.36e4i·7-s + 3.27e4i·8-s − 3.08e5·9-s + (−1.79e5 + 1.33e5i)10-s − 2.76e5·11-s − 7.13e5i·12-s − 7.26e5i·13-s − 2.35e6·14-s + (3.91e6 − 2.89e6i)15-s + 1.04e6·16-s + 3.21e6i·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + 1.65i·3-s − 0.5·4-s + (−0.594 − 0.803i)5-s + 1.17·6-s − 1.65i·7-s + 0.353i·8-s − 1.74·9-s + (−0.568 + 0.420i)10-s − 0.516·11-s − 0.828i·12-s − 0.543i·13-s − 1.17·14-s + (1.33 − 0.985i)15-s + 0.250·16-s + 0.548i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.803 + 0.594i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.803 + 0.594i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(10\)    =    \(2 \cdot 5\)
Sign: $-0.803 + 0.594i$
Analytic conductor: \(7.68343\)
Root analytic conductor: \(2.77190\)
Motivic weight: \(11\)
Rational: no
Arithmetic: yes
Character: $\chi_{10} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 10,\ (\ :11/2),\ -0.803 + 0.594i)\)

Particular Values

\(L(6)\) \(\approx\) \(0.176260 - 0.534495i\)
\(L(\frac12)\) \(\approx\) \(0.176260 - 0.534495i\)
\(L(\frac{13}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 32iT \)
5 \( 1 + (4.15e3 + 5.61e3i)T \)
good3 \( 1 - 697. iT - 1.77e5T^{2} \)
7 \( 1 + 7.36e4iT - 1.97e9T^{2} \)
11 \( 1 + 2.76e5T + 2.85e11T^{2} \)
13 \( 1 + 7.26e5iT - 1.79e12T^{2} \)
17 \( 1 - 3.21e6iT - 3.42e13T^{2} \)
19 \( 1 + 1.55e7T + 1.16e14T^{2} \)
23 \( 1 + 3.25e7iT - 9.52e14T^{2} \)
29 \( 1 - 4.53e7T + 1.22e16T^{2} \)
31 \( 1 + 1.06e8T + 2.54e16T^{2} \)
37 \( 1 - 1.65e8iT - 1.77e17T^{2} \)
41 \( 1 - 2.36e8T + 5.50e17T^{2} \)
43 \( 1 - 7.28e8iT - 9.29e17T^{2} \)
47 \( 1 + 1.41e9iT - 2.47e18T^{2} \)
53 \( 1 + 3.08e9iT - 9.26e18T^{2} \)
59 \( 1 - 7.61e9T + 3.01e19T^{2} \)
61 \( 1 - 1.45e9T + 4.35e19T^{2} \)
67 \( 1 + 1.32e10iT - 1.22e20T^{2} \)
71 \( 1 - 9.49e8T + 2.31e20T^{2} \)
73 \( 1 + 1.30e10iT - 3.13e20T^{2} \)
79 \( 1 + 5.30e10T + 7.47e20T^{2} \)
83 \( 1 - 2.15e10iT - 1.28e21T^{2} \)
89 \( 1 - 5.64e10T + 2.77e21T^{2} \)
97 \( 1 - 4.40e10iT - 7.15e21T^{2} \)
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

−17.17232521936282165785081236144, −16.29738944993287509508772026213, −14.79634159364417526234294717627, −12.98839272876659921220255413932, −10.92960987766972614129205590638, −10.14586142969081120788013892739, −8.400150554406774795262785234454, −4.69684669438466400053792838785, −3.77027776488121708317421053139, −0.28653568801652128920958238293, 2.38659173566895437058497228650, 5.95825738353647271869643663954, 7.26241876915768547059773176496, 8.606167586940627935121242023049, 11.64148641277401024043565559692, 12.81835837564538407152728669203, 14.40687303324101929481133598091, 15.65641800460853638570485636980, 17.74091208391727311162183871796, 18.68605628326089966007994562639

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