Properties

Label 2-300-4.3-c2-0-1
Degree $2$
Conductor $300$
Sign $-0.427 + 0.904i$
Analytic cond. $8.17440$
Root an. cond. $2.85909$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.438 + 1.95i)2-s + 1.73i·3-s + (−3.61 − 1.71i)4-s + (−3.37 − 0.758i)6-s + 6.33i·7-s + (4.92 − 6.30i)8-s − 2.99·9-s + 9.27i·11-s + (2.96 − 6.26i)12-s − 18.5·13-s + (−12.3 − 2.77i)14-s + (10.1 + 12.3i)16-s − 13.9·17-s + (1.31 − 5.85i)18-s − 17.2i·19-s + ⋯
L(s)  = 1  + (−0.219 + 0.975i)2-s + 0.577i·3-s + (−0.904 − 0.427i)4-s + (−0.563 − 0.126i)6-s + 0.904i·7-s + (0.615 − 0.788i)8-s − 0.333·9-s + 0.843i·11-s + (0.246 − 0.521i)12-s − 1.42·13-s + (−0.882 − 0.198i)14-s + (0.634 + 0.772i)16-s − 0.818·17-s + (0.0730 − 0.325i)18-s − 0.907i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.427 + 0.904i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.427 + 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $-0.427 + 0.904i$
Analytic conductor: \(8.17440\)
Root analytic conductor: \(2.85909\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{300} (151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 300,\ (\ :1),\ -0.427 + 0.904i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.228509 - 0.360844i\)
\(L(\frac12)\) \(\approx\) \(0.228509 - 0.360844i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.438 - 1.95i)T \)
3 \( 1 - 1.73iT \)
5 \( 1 \)
good7 \( 1 - 6.33iT - 49T^{2} \)
11 \( 1 - 9.27iT - 121T^{2} \)
13 \( 1 + 18.5T + 169T^{2} \)
17 \( 1 + 13.9T + 289T^{2} \)
19 \( 1 + 17.2iT - 361T^{2} \)
23 \( 1 + 33.7iT - 529T^{2} \)
29 \( 1 + 28.6T + 841T^{2} \)
31 \( 1 - 23.4iT - 961T^{2} \)
37 \( 1 - 67.3T + 1.36e3T^{2} \)
41 \( 1 + 44.0T + 1.68e3T^{2} \)
43 \( 1 - 50.2iT - 1.84e3T^{2} \)
47 \( 1 + 31.1iT - 2.20e3T^{2} \)
53 \( 1 + 81.6T + 2.80e3T^{2} \)
59 \( 1 - 19.2iT - 3.48e3T^{2} \)
61 \( 1 + 53.1T + 3.72e3T^{2} \)
67 \( 1 + 4.49iT - 4.48e3T^{2} \)
71 \( 1 - 13.3iT - 5.04e3T^{2} \)
73 \( 1 + 40.8T + 5.32e3T^{2} \)
79 \( 1 - 141. iT - 6.24e3T^{2} \)
83 \( 1 - 69.8iT - 6.88e3T^{2} \)
89 \( 1 + 46.3T + 7.92e3T^{2} \)
97 \( 1 + 68.5T + 9.40e3T^{2} \)
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

−12.30863177140148080935132199447, −11.05611975239189760218148302095, −9.889578961440088575575220814029, −9.308375210149818725674327902524, −8.430947144183682287075866435448, −7.28141684721059552365725853693, −6.33639482463952426618490988933, −5.05247840425779705689903546308, −4.48679034289566324898923396244, −2.49715562258898346522697079458, 0.21645978723584261123595078948, 1.78888308364050783113450319094, 3.22034456893542871142340455196, 4.41440114105526530880037629608, 5.77148683274426778693862535034, 7.33460498762057531257913450559, 7.939925116088796939972537658787, 9.209529875083343198101283726732, 10.00891887278397266440056906839, 11.05032408145075675823717121958

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