↑

Properties

Label 2-1045-5.4-c1-0-11
Degree $2$
Conductor $1045$
Sign $0.425 + 0.904i$
Analytic cond. $8.34436$
Root an. cond. $2.88866$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2.06i·2-s + 3.28i·3-s − 2.25·4-s + (0.951 + 2.02i)5-s − 6.78·6-s − 3.51i·7-s − 0.531i·8-s − 7.80·9-s + (−4.17 + 1.96i)10-s − 11-s − 7.42i·12-s + 6.22i·13-s + 7.26·14-s + (−6.65 + 3.12i)15-s − 3.41·16-s + 0.355i·17-s + ⋯
L(s)  = 1  + 1.45i·2-s + 1.89i·3-s − 1.12·4-s + (0.425 + 0.904i)5-s − 2.76·6-s − 1.33i·7-s − 0.188i·8-s − 2.60·9-s + (−1.32 + 0.620i)10-s − 0.301·11-s − 2.14i·12-s + 1.72i·13-s + 1.94·14-s + (−1.71 + 0.807i)15-s − 0.854·16-s + 0.0861i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.425 + 0.904i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.425 + 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $0.425 + 0.904i$
Analytic conductor: \(8.34436\)
Root analytic conductor: \(2.88866\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1045} (419, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1045,\ (\ :1/2),\ 0.425 + 0.904i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.278943092\)
\(L(\frac12)\) \(\approx\) \(1.278943092\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.951 - 2.02i)T \)
11 \( 1 + T \)
19 \( 1 - T \)
good2 \( 1 - 2.06iT - 2T^{2} \)
3 \( 1 - 3.28iT - 3T^{2} \)
7 \( 1 + 3.51iT - 7T^{2} \)
13 \( 1 - 6.22iT - 13T^{2} \)
17 \( 1 - 0.355iT - 17T^{2} \)
23 \( 1 + 2.52iT - 23T^{2} \)
29 \( 1 - 10.2T + 29T^{2} \)
31 \( 1 - 3.86T + 31T^{2} \)
37 \( 1 - 1.98iT - 37T^{2} \)
41 \( 1 + 3.07T + 41T^{2} \)
43 \( 1 + 1.96iT - 43T^{2} \)
47 \( 1 + 5.25iT - 47T^{2} \)
53 \( 1 - 6.40iT - 53T^{2} \)
59 \( 1 + 3.13T + 59T^{2} \)
61 \( 1 - 6.01T + 61T^{2} \)
67 \( 1 - 5.90iT - 67T^{2} \)
71 \( 1 - 7.16T + 71T^{2} \)
73 \( 1 - 16.2iT - 73T^{2} \)
79 \( 1 + 6.41T + 79T^{2} \)
83 \( 1 + 5.77iT - 83T^{2} \)
89 \( 1 + 12.1T + 89T^{2} \)
97 \( 1 + 0.839iT - 97T^{2} \)
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

−10.26657637569650302778024674255, −9.857559173398508855364417058805, −8.906664518697726442305373597426, −8.161190105014722140481357311481, −6.94177337865875558193061558758, −6.52755113214453325691488786267, −5.45686330753207916733048197463, −4.51513885847645599166008035691, −4.03685484826559302807631135409, −2.73412508818815398775888146162, 0.58091562373389650765867385670, 1.51982364267387543690785568741, 2.52715777728183229327479824493, 3.02853104314688688583386549492, 5.03965165939798980905913642394, 5.72723435784093949438673593494, 6.58941487098156847890949903462, 7.969918091595082983973373678454, 8.376549488681126613941555218460, 9.226255445006946153639475042449

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