↑

Properties

Label 2-1450-5.4-c1-0-29
Degree $2$
Conductor $1450$
Sign $0.447 + 0.894i$
Analytic cond. $11.5783$
Root an. cond. $3.40269$
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  + i·2-s − i·3-s − 4-s + 6-s − 4i·7-s − i·8-s + 2·9-s − 3·11-s + i·12-s + 4i·13-s + 4·14-s + 16-s + 3i·17-s + 2i·18-s + 7·19-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.577i·3-s − 0.5·4-s + 0.408·6-s − 1.51i·7-s − 0.353i·8-s + 0.666·9-s − 0.904·11-s + 0.288i·12-s + 1.10i·13-s + 1.06·14-s + 0.250·16-s + 0.727i·17-s + 0.471i·18-s + 1.60·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1450\)    =    \(2 \cdot 5^{2} \cdot 29\)
Sign: $0.447 + 0.894i$
Analytic conductor: \(11.5783\)
Root analytic conductor: \(3.40269\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1450} (349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1450,\ (\ :1/2),\ 0.447 + 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.410174012\)
\(L(\frac12)\) \(\approx\) \(1.410174012\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
5 \( 1 \)
29 \( 1 - T \)
good3 \( 1 + iT - 3T^{2} \)
7 \( 1 + 4iT - 7T^{2} \)
11 \( 1 + 3T + 11T^{2} \)
13 \( 1 - 4iT - 13T^{2} \)
17 \( 1 - 3iT - 17T^{2} \)
19 \( 1 - 7T + 19T^{2} \)
23 \( 1 + 6iT - 23T^{2} \)
31 \( 1 - 2T + 31T^{2} \)
37 \( 1 + 10iT - 37T^{2} \)
41 \( 1 + 9T + 41T^{2} \)
43 \( 1 + 8iT - 43T^{2} \)
47 \( 1 + 6iT - 47T^{2} \)
53 \( 1 + 12iT - 53T^{2} \)
59 \( 1 + 12T + 59T^{2} \)
61 \( 1 + 10T + 61T^{2} \)
67 \( 1 - 5iT - 67T^{2} \)
71 \( 1 - 6T + 71T^{2} \)
73 \( 1 - 7iT - 73T^{2} \)
79 \( 1 - 16T + 79T^{2} \)
83 \( 1 + 9iT - 83T^{2} \)
89 \( 1 - 3T + 89T^{2} \)
97 \( 1 - 2iT - 97T^{2} \)
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

−9.335790888368584240749618837736, −8.266714240331126175386647813062, −7.57281449864031005608553485860, −7.01624342080384826368112423821, −6.46092785256728271186354625123, −5.21264270280083631590115202246, −4.36795161578408285087638308539, −3.56240399268951005278088922693, −1.88571433587099519414305357489, −0.61108539328174925184052725970, 1.40673923246889987950456218687, 2.90134620037994912858659319079, 3.20071900515868833754496752282, 4.87513614562121279528226176641, 5.14862328618849141686170179274, 6.09224154178816486211386087185, 7.56992822796754940053980299156, 8.103378450349197800359869862916, 9.317204513104596479735639763960, 9.557101066917732012698362650386

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