↑

Properties

Label 2-405-45.7-c2-0-7
Degree $2$
Conductor $405$
Sign $-0.452 - 0.891i$
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  + (−2.67 + 0.718i)2-s + (3.20 − 1.84i)4-s + (4.08 − 2.88i)5-s + (−8.26 + 2.21i)7-s + (0.596 − 0.596i)8-s + (−8.87 + 10.6i)10-s + (−0.222 + 0.385i)11-s + (−3.63 − 0.975i)13-s + (20.5 − 11.8i)14-s + (−8.56 + 14.8i)16-s + (−14.8 − 14.8i)17-s + 29.1i·19-s + (7.75 − 16.7i)20-s + (0.319 − 1.19i)22-s + (19.8 + 5.32i)23-s + ⋯
L(s)  = 1  + (−1.33 + 0.359i)2-s + (0.800 − 0.462i)4-s + (0.817 − 0.576i)5-s + (−1.18 + 0.316i)7-s + (0.0745 − 0.0745i)8-s + (−0.887 + 1.06i)10-s + (−0.0202 + 0.0350i)11-s + (−0.279 − 0.0750i)13-s + (1.46 − 0.847i)14-s + (−0.535 + 0.926i)16-s + (−0.872 − 0.872i)17-s + 1.53i·19-s + (0.387 − 0.838i)20-s + (0.0145 − 0.0542i)22-s + (0.864 + 0.231i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.452 - 0.891i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.452 - 0.891i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(405\)    =    \(3^{4} \cdot 5\)
Sign: $-0.452 - 0.891i$
Analytic conductor: \(11.0354\)
Root analytic conductor: \(3.32196\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{405} (217, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 405,\ (\ :1),\ -0.452 - 0.891i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.246818 + 0.402125i\)
\(L(\frac12)\) \(\approx\) \(0.246818 + 0.402125i\)
\(L(2)\) 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 \( 1 + (-4.08 + 2.88i)T \)
good2 \( 1 + (2.67 - 0.718i)T + (3.46 - 2i)T^{2} \)
7 \( 1 + (8.26 - 2.21i)T + (42.4 - 24.5i)T^{2} \)
11 \( 1 + (0.222 - 0.385i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 + (3.63 + 0.975i)T + (146. + 84.5i)T^{2} \)
17 \( 1 + (14.8 + 14.8i)T + 289iT^{2} \)
19 \( 1 - 29.1iT - 361T^{2} \)
23 \( 1 + (-19.8 - 5.32i)T + (458. + 264.5i)T^{2} \)
29 \( 1 + (-34.7 - 20.0i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (7.64 + 13.2i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + (-6.61 - 6.61i)T + 1.36e3iT^{2} \)
41 \( 1 + (-21.2 - 36.7i)T + (-840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (-9.41 - 35.1i)T + (-1.60e3 + 924.5i)T^{2} \)
47 \( 1 + (72.3 - 19.3i)T + (1.91e3 - 1.10e3i)T^{2} \)
53 \( 1 + (44.8 - 44.8i)T - 2.80e3iT^{2} \)
59 \( 1 + (61.3 - 35.3i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (53.5 - 92.6i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (32.8 - 122. i)T + (-3.88e3 - 2.24e3i)T^{2} \)
71 \( 1 - 71.1T + 5.04e3T^{2} \)
73 \( 1 + (-77.9 + 77.9i)T - 5.32e3iT^{2} \)
79 \( 1 + (-58.2 - 33.6i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (-8.31 - 31.0i)T + (-5.96e3 + 3.44e3i)T^{2} \)
89 \( 1 + 104. iT - 7.92e3T^{2} \)
97 \( 1 + (143. - 38.5i)T + (8.14e3 - 4.70e3i)T^{2} \)
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

−10.93230790366769568088041124646, −9.930438804817976079933779033824, −9.499824446839485234814403854716, −8.824578041227159276022668121797, −7.82763781526259787302582865381, −6.69793488362392152850099993092, −6.01098348472307433585086228221, −4.62162503406398180605895624631, −2.83414270276775263521238819073, −1.26107704364591212916399410174, 0.34152612527800833167174253820, 2.07453387830017849307295789419, 3.12922753746309913538738427203, 4.90053117758006526848388553083, 6.51430513454362700402408907667, 6.89502344268642518759569726419, 8.239223850269277074309348091639, 9.334161968936256624468313343918, 9.601428571558301215151919397005, 10.76266643103974616523682092275

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