↑

Properties

Label 2-1200-5.3-c2-0-18
Degree $2$
Conductor $1200$
Sign $0.326 + 0.945i$
Analytic cond. $32.6976$
Root an. cond. $5.71818$
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  + (−1.22 − 1.22i)3-s + (−7.67 + 7.67i)7-s + 2.99i·9-s − 7.79·11-s + (−3.67 − 3.67i)13-s + (−3.34 + 3.34i)17-s + 28.3i·19-s + 18.7·21-s + (−14.4 − 14.4i)23-s + (3.67 − 3.67i)27-s − 43.3i·29-s + 40.7·31-s + (9.55 + 9.55i)33-s + (12.8 − 12.8i)37-s + 9i·39-s + ⋯
L(s)  = 1  + (−0.408 − 0.408i)3-s + (−1.09 + 1.09i)7-s + 0.333i·9-s − 0.708·11-s + (−0.282 − 0.282i)13-s + (−0.196 + 0.196i)17-s + 1.49i·19-s + 0.895·21-s + (−0.628 − 0.628i)23-s + (0.136 − 0.136i)27-s − 1.49i·29-s + 1.31·31-s + (0.289 + 0.289i)33-s + (0.348 − 0.348i)37-s + 0.230i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.326 + 0.945i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.326 + 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1200\)    =    \(2^{4} \cdot 3 \cdot 5^{2}\)
Sign: $0.326 + 0.945i$
Analytic conductor: \(32.6976\)
Root analytic conductor: \(5.71818\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1200} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1200,\ (\ :1),\ 0.326 + 0.945i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.7548417418\)
\(L(\frac12)\) \(\approx\) \(0.7548417418\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (1.22 + 1.22i)T \)
5 \( 1 \)
good7 \( 1 + (7.67 - 7.67i)T - 49iT^{2} \)
11 \( 1 + 7.79T + 121T^{2} \)
13 \( 1 + (3.67 + 3.67i)T + 169iT^{2} \)
17 \( 1 + (3.34 - 3.34i)T - 289iT^{2} \)
19 \( 1 - 28.3iT - 361T^{2} \)
23 \( 1 + (14.4 + 14.4i)T + 529iT^{2} \)
29 \( 1 + 43.3iT - 841T^{2} \)
31 \( 1 - 40.7T + 961T^{2} \)
37 \( 1 + (-12.8 + 12.8i)T - 1.36e3iT^{2} \)
41 \( 1 - 37.7T + 1.68e3T^{2} \)
43 \( 1 + (36.5 + 36.5i)T + 1.84e3iT^{2} \)
47 \( 1 + (52.2 - 52.2i)T - 2.20e3iT^{2} \)
53 \( 1 + (-47.1 - 47.1i)T + 2.80e3iT^{2} \)
59 \( 1 + 82iT - 3.48e3T^{2} \)
61 \( 1 - 55.4T + 3.72e3T^{2} \)
67 \( 1 + (-59.2 + 59.2i)T - 4.48e3iT^{2} \)
71 \( 1 - 58.2T + 5.04e3T^{2} \)
73 \( 1 + (-21.7 - 21.7i)T + 5.32e3iT^{2} \)
79 \( 1 - 91.9iT - 6.24e3T^{2} \)
83 \( 1 + (57.1 + 57.1i)T + 6.88e3iT^{2} \)
89 \( 1 + 120. iT - 7.92e3T^{2} \)
97 \( 1 + (103. - 103. i)T - 9.40e3iT^{2} \)
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

−9.673463394669237908930090184315, −8.338518426361826527201513827815, −7.951431522989933796128622494795, −6.67280993970042838402645348450, −6.03119454288081281747690953739, −5.44654856292737170112060758343, −4.18258523079743429189015388735, −2.91675076769961110135853707930, −2.10528376377791410817311469738, −0.32397315696579974703855549052, 0.804139589648096137493068581322, 2.65387740576460835410311137362, 3.60896042948492696472342169615, 4.57582982066390531456740611577, 5.37152177391065093461753563281, 6.64904589432122808647540687937, 6.94069196070718676620393940186, 8.041777574146976160231618449975, 9.122758586279031818696949929515, 9.891196596974555222192664584604

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