↑

Properties

Label 2-405-45.13-c2-0-2
Degree $2$
Conductor $405$
Sign $-0.614 - 0.788i$
Analytic cond. $11.0354$
Root an. cond. $3.32196$
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  + (0.933 + 0.250i)2-s + (−2.65 − 1.53i)4-s + (−2.47 − 4.34i)5-s + (−2.61 − 0.700i)7-s + (−4.82 − 4.82i)8-s + (−1.22 − 4.67i)10-s + (8.02 + 13.8i)11-s + (−0.830 + 0.222i)13-s + (−2.26 − 1.30i)14-s + (2.83 + 4.90i)16-s + (−7.56 + 7.56i)17-s + 21.0i·19-s + (−0.0990 + 15.3i)20-s + (4.01 + 14.9i)22-s + (−34.5 + 9.25i)23-s + ⋯
L(s)  = 1  + (0.466 + 0.125i)2-s + (−0.663 − 0.383i)4-s + (−0.494 − 0.869i)5-s + (−0.373 − 0.100i)7-s + (−0.603 − 0.603i)8-s + (−0.122 − 0.467i)10-s + (0.729 + 1.26i)11-s + (−0.0638 + 0.0171i)13-s + (−0.161 − 0.0933i)14-s + (0.177 + 0.306i)16-s + (−0.444 + 0.444i)17-s + 1.10i·19-s + (−0.00495 + 0.766i)20-s + (0.182 + 0.680i)22-s + (−1.50 + 0.402i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.614 - 0.788i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.614 - 0.788i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(405\)    =    \(3^{4} \cdot 5\)
Sign: $-0.614 - 0.788i$
Analytic conductor: \(11.0354\)
Root analytic conductor: \(3.32196\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{405} (28, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 405,\ (\ :1),\ -0.614 - 0.788i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.168085 + 0.343956i\)
\(L(\frac12)\) \(\approx\) \(0.168085 + 0.343956i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + (2.47 + 4.34i)T \)
good2 \( 1 + (-0.933 - 0.250i)T + (3.46 + 2i)T^{2} \)
7 \( 1 + (2.61 + 0.700i)T + (42.4 + 24.5i)T^{2} \)
11 \( 1 + (-8.02 - 13.8i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + (0.830 - 0.222i)T + (146. - 84.5i)T^{2} \)
17 \( 1 + (7.56 - 7.56i)T - 289iT^{2} \)
19 \( 1 - 21.0iT - 361T^{2} \)
23 \( 1 + (34.5 - 9.25i)T + (458. - 264.5i)T^{2} \)
29 \( 1 + (11.6 - 6.72i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + (-19.6 + 33.9i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + (-5.47 + 5.47i)T - 1.36e3iT^{2} \)
41 \( 1 + (34.7 - 60.1i)T + (-840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-20.3 + 75.9i)T + (-1.60e3 - 924.5i)T^{2} \)
47 \( 1 + (18.7 + 5.01i)T + (1.91e3 + 1.10e3i)T^{2} \)
53 \( 1 + (-34.8 - 34.8i)T + 2.80e3iT^{2} \)
59 \( 1 + (53.6 + 31.0i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-33.6 - 58.2i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-10.7 - 40.0i)T + (-3.88e3 + 2.24e3i)T^{2} \)
71 \( 1 + 71.3T + 5.04e3T^{2} \)
73 \( 1 + (23.5 + 23.5i)T + 5.32e3iT^{2} \)
79 \( 1 + (-4.86 + 2.80i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (9.55 - 35.6i)T + (-5.96e3 - 3.44e3i)T^{2} \)
89 \( 1 - 23.2iT - 7.92e3T^{2} \)
97 \( 1 + (132. + 35.6i)T + (8.14e3 + 4.70e3i)T^{2} \)
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

−11.74153814523066044738050992747, −10.12302389001825149100112470714, −9.636220371332069900994486971564, −8.683020769959025393567586570103, −7.71107728562678908342655395961, −6.44802824682803448893697523074, −5.49173686027280635272301696634, −4.31183371094244570428125911014, −3.87178820745627572721506973630, −1.60368221393429584455624404370, 0.14584795193178123336072118856, 2.76412628048784463153703112063, 3.58487540388353634971997262202, 4.57875367820555523322128661983, 5.95892696590286709463925393831, 6.80443801537564399775571322774, 8.051423479702546998352762599488, 8.815360686353882334246938476418, 9.775281906214477407162537440187, 10.98974890563753982143913275783

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