Properties

Label 2-1755-195.194-c0-0-8
Degree $2$
Conductor $1755$
Sign $-0.965 + 0.258i$
Analytic cond. $0.875859$
Root an. cond. $0.935873$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 1.93i·2-s − 2.73·4-s + (0.965 − 0.258i)5-s + 3.34i·8-s + (−0.499 − 1.86i)10-s + 1.41·11-s i·13-s + 3.73·16-s + (−2.63 + 0.707i)20-s − 2.73i·22-s + (0.866 − 0.499i)25-s − 1.93·26-s − 3.86i·32-s + (0.866 + 3.23i)40-s − 1.41·41-s + ⋯
L(s)  = 1  − 1.93i·2-s − 2.73·4-s + (0.965 − 0.258i)5-s + 3.34i·8-s + (−0.499 − 1.86i)10-s + 1.41·11-s i·13-s + 3.73·16-s + (−2.63 + 0.707i)20-s − 2.73i·22-s + (0.866 − 0.499i)25-s − 1.93·26-s − 3.86i·32-s + (0.866 + 3.23i)40-s − 1.41·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1755 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.965 + 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1755 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.965 + 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1755\)    =    \(3^{3} \cdot 5 \cdot 13\)
Sign: $-0.965 + 0.258i$
Analytic conductor: \(0.875859\)
Root analytic conductor: \(0.935873\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1755} (1754, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1755,\ (\ :0),\ -0.965 + 0.258i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.177264762\)
\(L(\frac12)\) \(\approx\) \(1.177264762\)
\(L(1)\) 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 \( 1 + (-0.965 + 0.258i)T \)
13 \( 1 + iT \)
good2 \( 1 + 1.93iT - T^{2} \)
7 \( 1 + T^{2} \)
11 \( 1 - 1.41T + T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 + 1.41T + T^{2} \)
43 \( 1 + 1.73iT - T^{2} \)
47 \( 1 - 1.93iT - T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 - 0.517T + T^{2} \)
61 \( 1 + 1.73T + T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 - 1.93T + T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 - 0.517iT - T^{2} \)
89 \( 1 + 0.517T + T^{2} \)
97 \( 1 + T^{2} \)
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

−9.360630716139457539285365691020, −8.851351417694123452518069325753, −8.012152949107694537858635274353, −6.53261757533318610897836169732, −5.53978494956910288631850481955, −4.78931398198486691118932562157, −3.81199455499362245468869474573, −2.99019089290238840585139406709, −1.93602750504928710509457078007, −1.07954610678329079796729128208, 1.56455261832489442377760691537, 3.48757874884375130590127918937, 4.44994430759441116589450016583, 5.21861194227029662598380458676, 6.19013854164478768875014620992, 6.59606354463172542856718045498, 7.14494406012842477350987727912, 8.255823291563823724526134692438, 8.945955446025123580555121922601, 9.509592104828348599514899249018

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