↑

Properties

Label 2-950-19.7-c1-0-29
Degree $2$
Conductor $950$
Sign $-0.813 - 0.582i$
Analytic cond. $7.58578$
Root an. cond. $2.75423$
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 + (−0.895 − 1.55i)3-s + (−0.499 + 0.866i)4-s + (−0.895 + 1.55i)6-s − 7-s + 0.999·8-s + (−0.104 + 0.180i)9-s + 0.791·11-s + 1.79·12-s + (2.39 − 4.14i)13-s + (0.5 + 0.866i)14-s + (−0.5 − 0.866i)16-s + (−1.89 − 3.28i)17-s + 0.208·18-s + (−3.5 + 2.59i)19-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.517 − 0.895i)3-s + (−0.249 + 0.433i)4-s + (−0.365 + 0.633i)6-s − 0.377·7-s + 0.353·8-s + (−0.0347 + 0.0602i)9-s + 0.238·11-s + 0.517·12-s + (0.664 − 1.15i)13-s + (0.133 + 0.231i)14-s + (−0.125 − 0.216i)16-s + (−0.459 − 0.796i)17-s + 0.0491·18-s + (−0.802 + 0.596i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.813 - 0.582i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.813 - 0.582i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $-0.813 - 0.582i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (501, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ -0.813 - 0.582i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.168588 + 0.524953i\)
\(L(\frac12)\) \(\approx\) \(0.168588 + 0.524953i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 + 0.866i)T \)
5 \( 1 \)
19 \( 1 + (3.5 - 2.59i)T \)
good3 \( 1 + (0.895 + 1.55i)T + (-1.5 + 2.59i)T^{2} \)
7 \( 1 + T + 7T^{2} \)
11 \( 1 - 0.791T + 11T^{2} \)
13 \( 1 + (-2.39 + 4.14i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (1.89 + 3.28i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (-2.29 + 3.96i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.10 - 1.91i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 0.208T + 31T^{2} \)
37 \( 1 + 5.58T + 37T^{2} \)
41 \( 1 + (1.18 + 2.05i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.604 + 1.04i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (3.08 - 5.33i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (2.29 - 3.96i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2.29 - 3.96i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6.18 + 10.7i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.20 - 5.55i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (2.29 + 3.96i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-0.895 - 1.55i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-3.97 - 6.89i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 0.791T + 83T^{2} \)
89 \( 1 + (-2.29 + 3.96i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (0.686 + 1.18i)T + (-48.5 + 84.0i)T^{2} \)
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

−9.606776123673394391910524258422, −8.713952772701391866958467222466, −7.923517405490934025502151229856, −6.92919663027319495551193921645, −6.29708200579436353382028120313, −5.24151446276823908118968629732, −3.92150115171981755146453219793, −2.86013667616916050672901539336, −1.52598860025458998598233316743, −0.31739915112750675840137392031, 1.80027901295484106843826386505, 3.68671973153218491119915866613, 4.45328141117136098616565828147, 5.36762872513775232811407825336, 6.36385179773032175085827940586, 6.92619657270344543160360310738, 8.170698199691514174807651742542, 9.006401776919535664742955738616, 9.600801142868577471345258557826, 10.47652931607654735059428757645

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