↑

Properties

Label 2-950-25.16-c1-0-28
Degree $2$
Conductor $950$
Sign $-0.0603 - 0.998i$
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 + (0.611 + 1.88i)3-s + (0.309 + 0.951i)4-s + (2.16 − 0.553i)5-s + (−0.611 + 1.88i)6-s + 2.35·7-s + (−0.309 + 0.951i)8-s + (−0.738 + 0.536i)9-s + (2.07 + 0.825i)10-s + (−2.31 − 1.68i)11-s + (−1.60 + 1.16i)12-s + (−0.514 + 0.373i)13-s + (1.90 + 1.38i)14-s + (2.36 + 3.73i)15-s + (−0.809 + 0.587i)16-s + (−1.52 + 4.69i)17-s + ⋯
L(s)  = 1  + (0.572 + 0.415i)2-s + (0.352 + 1.08i)3-s + (0.154 + 0.475i)4-s + (0.968 − 0.247i)5-s + (−0.249 + 0.767i)6-s + 0.888·7-s + (−0.109 + 0.336i)8-s + (−0.246 + 0.178i)9-s + (0.657 + 0.261i)10-s + (−0.698 − 0.507i)11-s + (−0.461 + 0.335i)12-s + (−0.142 + 0.103i)13-s + (0.508 + 0.369i)14-s + (0.610 + 0.964i)15-s + (−0.202 + 0.146i)16-s + (−0.369 + 1.13i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0603 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0603 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $-0.0603 - 0.998i$
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: \((2,\ 950,\ (\ :1/2),\ -0.0603 - 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.09374 + 2.22422i\)
\(L(\frac12)\) \(\approx\) \(2.09374 + 2.22422i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.16 + 0.553i)T \)
19 \( 1 + (0.309 - 0.951i)T \)
good3 \( 1 + (-0.611 - 1.88i)T + (-2.42 + 1.76i)T^{2} \)
7 \( 1 - 2.35T + 7T^{2} \)
11 \( 1 + (2.31 + 1.68i)T + (3.39 + 10.4i)T^{2} \)
13 \( 1 + (0.514 - 0.373i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (1.52 - 4.69i)T + (-13.7 - 9.99i)T^{2} \)
23 \( 1 + (-1.67 - 1.21i)T + (7.10 + 21.8i)T^{2} \)
29 \( 1 + (0.148 + 0.455i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-3.13 + 9.65i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-3.88 + 2.81i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (7.46 - 5.42i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 11.6T + 43T^{2} \)
47 \( 1 + (-0.509 - 1.56i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (0.610 + 1.87i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-7.34 + 5.33i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (4.78 + 3.47i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + (3.85 - 11.8i)T + (-54.2 - 39.3i)T^{2} \)
71 \( 1 + (2.82 + 8.68i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-7.07 - 5.13i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (4.04 + 12.4i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-2.07 + 6.39i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + (-8.47 - 6.15i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (-0.486 - 1.49i)T + (-78.4 + 57.0i)T^{2} \)
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

−10.18335609860441963361217538606, −9.480225836182401686893331519227, −8.494528181546298442215734461988, −8.002245634830723048169281110322, −6.60956875839901437109741847470, −5.72471817111416470656798815624, −4.93059275679138836785444859157, −4.25195390806438452805716656532, −3.11468344127250973279616329489, −1.86133156445796496235411117723, 1.33469596700295303048417624384, 2.22456158593273556245166075476, 2.97202760136313515909926838176, 4.82134502827293373670426377704, 5.19636068824849208026412687794, 6.60987351043837278891219807642, 7.05295651721104972682927172465, 8.064036387873761349006076944700, 8.951280905138978548350054592719, 10.05726449168751370869782690630

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