↑

Properties

Label 2-1254-209.65-c1-0-17
Degree $2$
Conductor $1254$
Sign $0.889 - 0.456i$
Analytic cond. $10.0132$
Root an. cond. $3.16437$
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  + (0.5 − 0.866i)2-s + (0.866 + 0.5i)3-s + (−0.499 − 0.866i)4-s + (−0.681 + 1.18i)5-s + (0.866 − 0.499i)6-s − 0.440i·7-s − 0.999·8-s + (0.499 + 0.866i)9-s + (0.681 + 1.18i)10-s + (2.75 + 1.85i)11-s − 0.999i·12-s + (2.12 + 3.67i)13-s + (−0.381 − 0.220i)14-s + (−1.18 + 0.681i)15-s + (−0.5 + 0.866i)16-s + (−1.61 − 0.930i)17-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (0.499 + 0.288i)3-s + (−0.249 − 0.433i)4-s + (−0.304 + 0.528i)5-s + (0.353 − 0.204i)6-s − 0.166i·7-s − 0.353·8-s + (0.166 + 0.288i)9-s + (0.215 + 0.373i)10-s + (0.829 + 0.557i)11-s − 0.288i·12-s + (0.588 + 1.02i)13-s + (−0.101 − 0.0588i)14-s + (−0.304 + 0.176i)15-s + (−0.125 + 0.216i)16-s + (−0.390 − 0.225i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1254 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.889 - 0.456i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1254 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.889 - 0.456i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1254\)    =    \(2 \cdot 3 \cdot 11 \cdot 19\)
Sign: $0.889 - 0.456i$
Analytic conductor: \(10.0132\)
Root analytic conductor: \(3.16437\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1254} (901, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1254,\ (\ :1/2),\ 0.889 - 0.456i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.149740373\)
\(L(\frac12)\) \(\approx\) \(2.149740373\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 + 0.866i)T \)
3 \( 1 + (-0.866 - 0.5i)T \)
11 \( 1 + (-2.75 - 1.85i)T \)
19 \( 1 + (4.34 + 0.344i)T \)
good5 \( 1 + (0.681 - 1.18i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + 0.440iT - 7T^{2} \)
13 \( 1 + (-2.12 - 3.67i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.61 + 0.930i)T + (8.5 + 14.7i)T^{2} \)
23 \( 1 + (-3.53 - 6.12i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.67 + 4.63i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 0.0632iT - 31T^{2} \)
37 \( 1 - 7.40iT - 37T^{2} \)
41 \( 1 + (-0.0722 + 0.125i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-7.46 - 4.31i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (6.08 + 10.5i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2.86 + 1.65i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-11.3 - 6.57i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-12.1 + 7.04i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (7.77 - 4.48i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-6.03 - 3.48i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (0.848 + 0.489i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.259 + 0.449i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 0.315iT - 83T^{2} \)
89 \( 1 + (8.87 - 5.12i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (0.113 + 0.0657i)T + (48.5 + 84.0i)T^{2} \)
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

−9.708682097117252926957296074556, −9.152781696629962285566129357612, −8.338392852145221087625096111718, −7.10122340471865327738987287865, −6.62181339708858381313712170411, −5.33369090467911615678039514981, −4.15660027351611044379613009533, −3.81108087406114797867289046823, −2.59590655151706720214309533632, −1.51976944621083041946145969967, 0.823608547775802055471086840518, 2.50666443373098551183830154313, 3.67217617188036042638541473922, 4.39171229354510910606476767381, 5.53882616685595990829912411457, 6.34625048492379781132873220237, 7.11008460298782158895421130783, 8.180178806307375713943482759506, 8.652474746068101749606714161172, 9.143488329865804608736310537511

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