↑

Properties

Label 2-950-19.5-c1-0-26
Degree $2$
Conductor $950$
Sign $0.388 + 0.921i$
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  + (0.939 − 0.342i)2-s + (−0.501 + 2.84i)3-s + (0.766 − 0.642i)4-s + (0.501 + 2.84i)6-s + (−2.01 − 3.49i)7-s + (0.500 − 0.866i)8-s + (−5.02 − 1.82i)9-s + (2.11 − 3.65i)11-s + (1.44 + 2.50i)12-s + (−0.959 − 5.44i)13-s + (−3.09 − 2.59i)14-s + (0.173 − 0.984i)16-s + (−1.43 + 0.523i)17-s − 5.34·18-s + (−0.805 − 4.28i)19-s + ⋯
L(s)  = 1  + (0.664 − 0.241i)2-s + (−0.289 + 1.64i)3-s + (0.383 − 0.321i)4-s + (0.204 + 1.16i)6-s + (−0.762 − 1.32i)7-s + (0.176 − 0.306i)8-s + (−1.67 − 0.609i)9-s + (0.636 − 1.10i)11-s + (0.417 + 0.722i)12-s + (−0.266 − 1.50i)13-s + (−0.826 − 0.693i)14-s + (0.0434 − 0.246i)16-s + (−0.349 + 0.127i)17-s − 1.26·18-s + (−0.184 − 0.982i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.388 + 0.921i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.388 + 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $0.388 + 0.921i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (651, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ 0.388 + 0.921i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.19202 - 0.790650i\)
\(L(\frac12)\) \(\approx\) \(1.19202 - 0.790650i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.939 + 0.342i)T \)
5 \( 1 \)
19 \( 1 + (0.805 + 4.28i)T \)
good3 \( 1 + (0.501 - 2.84i)T + (-2.81 - 1.02i)T^{2} \)
7 \( 1 + (2.01 + 3.49i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.11 + 3.65i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.959 + 5.44i)T + (-12.2 + 4.44i)T^{2} \)
17 \( 1 + (1.43 - 0.523i)T + (13.0 - 10.9i)T^{2} \)
23 \( 1 + (4.57 - 3.83i)T + (3.99 - 22.6i)T^{2} \)
29 \( 1 + (-4.97 - 1.81i)T + (22.2 + 18.6i)T^{2} \)
31 \( 1 + (1.35 + 2.35i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 8.71T + 37T^{2} \)
41 \( 1 + (0.560 - 3.17i)T + (-38.5 - 14.0i)T^{2} \)
43 \( 1 + (1.74 + 1.46i)T + (7.46 + 42.3i)T^{2} \)
47 \( 1 + (-9.93 - 3.61i)T + (36.0 + 30.2i)T^{2} \)
53 \( 1 + (1.97 - 1.66i)T + (9.20 - 52.1i)T^{2} \)
59 \( 1 + (-9.90 + 3.60i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (3.45 - 2.89i)T + (10.5 - 60.0i)T^{2} \)
67 \( 1 + (-10.3 - 3.76i)T + (51.3 + 43.0i)T^{2} \)
71 \( 1 + (5.73 + 4.81i)T + (12.3 + 69.9i)T^{2} \)
73 \( 1 + (0.342 - 1.94i)T + (-68.5 - 24.9i)T^{2} \)
79 \( 1 + (-2.15 + 12.2i)T + (-74.2 - 27.0i)T^{2} \)
83 \( 1 + (-1.91 - 3.32i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-1.96 - 11.1i)T + (-83.6 + 30.4i)T^{2} \)
97 \( 1 + (-12.8 + 4.66i)T + (74.3 - 62.3i)T^{2} \)
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

−10.16565003581672373649127121781, −9.408982856153082417393837229194, −8.412502622895241622452256581761, −7.13448072076824789992564089793, −6.11898773867006333921590419752, −5.35531818197331962807547577608, −4.37106116314074626997929378062, −3.63013078503804172411133475118, −3.06681221203805298395538899538, −0.52953453146693117374491058768, 1.88934957669426690185433469498, 2.36894208263455844438128614235, 3.96430023881916438471927963684, 5.20481578157957106759070216689, 6.26393271847689205046137103284, 6.58893286482674293870073636451, 7.28273756105959071616420993685, 8.405110774072920216857226308287, 9.106056453507565294626409720118, 10.20816446170614190398875139749

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