↑

Properties

Label 2-1755-195.194-c0-0-1
Degree $2$
Conductor $1755$
Sign $-0.258 - 0.965i$
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  + 0.517i·2-s + 0.732·4-s + (0.258 + 0.965i)5-s + 0.896i·8-s + (−0.499 + 0.133i)10-s − 1.41·11-s + i·13-s + 0.267·16-s + (0.189 + 0.707i)20-s − 0.732i·22-s + (−0.866 + 0.499i)25-s − 0.517·26-s + 1.03i·32-s + (−0.866 + 0.232i)40-s + 1.41·41-s + ⋯
L(s)  = 1  + 0.517i·2-s + 0.732·4-s + (0.258 + 0.965i)5-s + 0.896i·8-s + (−0.499 + 0.133i)10-s − 1.41·11-s + i·13-s + 0.267·16-s + (0.189 + 0.707i)20-s − 0.732i·22-s + (−0.866 + 0.499i)25-s − 0.517·26-s + 1.03i·32-s + (−0.866 + 0.232i)40-s + 1.41·41-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1755\)    =    \(3^{3} \cdot 5 \cdot 13\)
Sign: $-0.258 - 0.965i$
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: \((2,\ 1755,\ (\ :0),\ -0.258 - 0.965i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.309262859\)
\(L(\frac12)\) \(\approx\) \(1.309262859\)
\(L(1)\) 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 \( 1 + (-0.258 - 0.965i)T \)
13 \( 1 - iT \)
good2 \( 1 - 0.517iT - T^{2} \)
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 \( 1 + 0.517iT - T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 - 1.93T + T^{2} \)
61 \( 1 - 1.73T + T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 - 0.517T + T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 + 1.93iT - T^{2} \)
89 \( 1 + 1.93T + T^{2} \)
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

−9.917008741039699525776081972219, −8.773216212051696635432625406917, −7.914156176888338809228083641764, −7.21405561104399317727501107243, −6.69092319048366570273636181455, −5.81166699762107222186723298027, −5.13846012610212564716215653285, −3.77892985942341279739723313472, −2.64428558071648828393615886469, −2.05504874907770580330091101057, 0.976448102218952870420147893862, 2.27779873847248769813557799926, 3.02629467545886330389109550772, 4.23514220744533207148480075528, 5.30161297233946095857483222204, 5.82997126261432243167467387023, 6.92199678449886022589618317978, 7.943794419092831860779044482375, 8.253661030158286043529094281423, 9.587445634514543964286075666879

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