↑

Properties

Label 2-950-19.9-c1-0-16
Degree $2$
Conductor $950$
Sign $0.533 - 0.845i$
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  + (0.173 + 0.984i)2-s + (1.28 − 1.07i)3-s + (−0.939 + 0.342i)4-s + (1.28 + 1.07i)6-s + (0.237 − 0.411i)7-s + (−0.5 − 0.866i)8-s + (−0.0320 + 0.181i)9-s + (2.79 + 4.84i)11-s + (−0.838 + 1.45i)12-s + (−1.88 − 1.58i)13-s + (0.446 + 0.162i)14-s + (0.766 − 0.642i)16-s + (0.498 + 2.82i)17-s − 0.184·18-s + (3.17 + 2.98i)19-s + ⋯
L(s)  = 1  + (0.122 + 0.696i)2-s + (0.742 − 0.622i)3-s + (−0.469 + 0.171i)4-s + (0.524 + 0.440i)6-s + (0.0898 − 0.155i)7-s + (−0.176 − 0.306i)8-s + (−0.0106 + 0.0606i)9-s + (0.843 + 1.46i)11-s + (−0.242 + 0.419i)12-s + (−0.523 − 0.438i)13-s + (0.119 + 0.0434i)14-s + (0.191 − 0.160i)16-s + (0.121 + 0.686i)17-s − 0.0435·18-s + (0.729 + 0.683i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.533 - 0.845i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.533 - 0.845i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $0.533 - 0.845i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (351, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ 0.533 - 0.845i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.86374 + 1.02784i\)
\(L(\frac12)\) \(\approx\) \(1.86374 + 1.02784i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.173 - 0.984i)T \)
5 \( 1 \)
19 \( 1 + (-3.17 - 2.98i)T \)
good3 \( 1 + (-1.28 + 1.07i)T + (0.520 - 2.95i)T^{2} \)
7 \( 1 + (-0.237 + 0.411i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-2.79 - 4.84i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.88 + 1.58i)T + (2.25 + 12.8i)T^{2} \)
17 \( 1 + (-0.498 - 2.82i)T + (-15.9 + 5.81i)T^{2} \)
23 \( 1 + (-4.12 + 1.50i)T + (17.6 - 14.7i)T^{2} \)
29 \( 1 + (-1.51 + 8.59i)T + (-27.2 - 9.91i)T^{2} \)
31 \( 1 + (2.88 - 4.99i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 6.02T + 37T^{2} \)
41 \( 1 + (-5.18 + 4.34i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 + (-0.433 - 0.157i)T + (32.9 + 27.6i)T^{2} \)
47 \( 1 + (-0.456 + 2.59i)T + (-44.1 - 16.0i)T^{2} \)
53 \( 1 + (7.43 - 2.70i)T + (40.6 - 34.0i)T^{2} \)
59 \( 1 + (1.42 + 8.06i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (2.98 - 1.08i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (0.0262 - 0.148i)T + (-62.9 - 22.9i)T^{2} \)
71 \( 1 + (4.48 + 1.63i)T + (54.3 + 45.6i)T^{2} \)
73 \( 1 + (7.04 - 5.91i)T + (12.6 - 71.8i)T^{2} \)
79 \( 1 + (-9.48 + 7.96i)T + (13.7 - 77.7i)T^{2} \)
83 \( 1 + (6.59 - 11.4i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (1.23 + 1.03i)T + (15.4 + 87.6i)T^{2} \)
97 \( 1 + (0.873 + 4.95i)T + (-91.1 + 33.1i)T^{2} \)
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

−9.870199327059711225640105604004, −9.239779397061291997299509967176, −8.236001418760688128880892490030, −7.56980114273584130589246411395, −7.06031261586351005623584308921, −6.03016879250686875721312688272, −4.90885083392617781023320062615, −4.01712329750533566481744696342, −2.71442258197814918729088589971, −1.48888947219484950749194933336, 1.01868266962988604268056867175, 2.75819938596764378565323232547, 3.33362500873495280323779117236, 4.32586573830349689346487579501, 5.30383667483258978675419443439, 6.40017363518695581892195956880, 7.53822087018267815853406350450, 8.714087659834392463322317589011, 9.185866025502806505619229682685, 9.646170552450668513041344314065

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