↑

Properties

Label 2-950-25.16-c1-0-31
Degree $2$
Conductor $950$
Sign $0.852 - 0.521i$
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  + (0.809 + 0.587i)2-s + (−0.183 − 0.565i)3-s + (0.309 + 0.951i)4-s + (2.14 + 0.641i)5-s + (0.183 − 0.565i)6-s + 2.23·7-s + (−0.309 + 0.951i)8-s + (2.14 − 1.55i)9-s + (1.35 + 1.77i)10-s + (1.48 + 1.07i)11-s + (0.481 − 0.349i)12-s + (−3.23 + 2.35i)13-s + (1.81 + 1.31i)14-s + (−0.0309 − 1.33i)15-s + (−0.809 + 0.587i)16-s + (1.44 − 4.44i)17-s + ⋯
L(s)  = 1  + (0.572 + 0.415i)2-s + (−0.106 − 0.326i)3-s + (0.154 + 0.475i)4-s + (0.957 + 0.286i)5-s + (0.0750 − 0.231i)6-s + 0.845·7-s + (−0.109 + 0.336i)8-s + (0.713 − 0.518i)9-s + (0.428 + 0.562i)10-s + (0.447 + 0.325i)11-s + (0.138 − 0.100i)12-s + (−0.897 + 0.652i)13-s + (0.483 + 0.351i)14-s + (−0.00799 − 0.343i)15-s + (−0.202 + 0.146i)16-s + (0.350 − 1.07i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.852 - 0.521i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.852 - 0.521i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $0.852 - 0.521i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (191, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ 0.852 - 0.521i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.69833 + 0.760134i\)
\(L(\frac12)\) \(\approx\) \(2.69833 + 0.760134i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.809 - 0.587i)T \)
5 \( 1 + (-2.14 - 0.641i)T \)
19 \( 1 + (0.309 - 0.951i)T \)
good3 \( 1 + (0.183 + 0.565i)T + (-2.42 + 1.76i)T^{2} \)
7 \( 1 - 2.23T + 7T^{2} \)
11 \( 1 + (-1.48 - 1.07i)T + (3.39 + 10.4i)T^{2} \)
13 \( 1 + (3.23 - 2.35i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-1.44 + 4.44i)T + (-13.7 - 9.99i)T^{2} \)
23 \( 1 + (3.11 + 2.26i)T + (7.10 + 21.8i)T^{2} \)
29 \( 1 + (0.690 + 2.12i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (1.86 - 5.72i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-2.58 + 1.88i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (1.30 - 0.948i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 1.82T + 43T^{2} \)
47 \( 1 + (0.959 + 2.95i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-1.91 - 5.88i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-5.27 + 3.83i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-2.81 - 2.04i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + (-0.189 + 0.583i)T + (-54.2 - 39.3i)T^{2} \)
71 \( 1 + (-3.19 - 9.83i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (2.22 + 1.61i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (-0.932 - 2.87i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (4.97 - 15.3i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + (7.50 + 5.45i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (3.83 + 11.8i)T + (-78.4 + 57.0i)T^{2} \)
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

−9.907493673022344990913381335466, −9.443867497078374021290125010315, −8.296803902424983967522842691321, −7.14500028352637657856455448148, −6.87803947102069582984310036140, −5.79052653071627498620927692301, −4.92243799674003527804169811868, −4.06933373449020918400390109822, −2.56850075055032274363493377454, −1.54481240743360394269776017116, 1.41814836877082652624548022288, 2.31037374393506991782431977566, 3.78361824423740470402378354066, 4.75013896445369340184387890997, 5.40706922583358727584784016017, 6.21711860065412766889178053443, 7.45200057787076010937320283752, 8.319685460904858353454576318964, 9.461862273687235028107087277492, 10.07569948424646068921430660402

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