↑

Properties

Label 2-950-95.12-c1-0-26
Degree $2$
Conductor $950$
Sign $0.426 + 0.904i$
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  + (0.965 + 0.258i)2-s + (0.824 − 3.07i)3-s + (0.866 + 0.499i)4-s + (1.59 − 2.75i)6-s + (3.29 + 3.29i)7-s + (0.707 + 0.707i)8-s + (−6.18 − 3.56i)9-s + 1.07·11-s + (2.25 − 2.25i)12-s + (−0.991 + 0.265i)13-s + (2.32 + 4.03i)14-s + (0.500 + 0.866i)16-s + (1.62 − 6.05i)17-s + (−5.04 − 5.04i)18-s + (3.41 − 2.70i)19-s + ⋯
L(s)  = 1  + (0.683 + 0.183i)2-s + (0.475 − 1.77i)3-s + (0.433 + 0.249i)4-s + (0.649 − 1.12i)6-s + (1.24 + 1.24i)7-s + (0.249 + 0.249i)8-s + (−2.06 − 1.18i)9-s + 0.323·11-s + (0.649 − 0.649i)12-s + (−0.275 + 0.0737i)13-s + (0.622 + 1.07i)14-s + (0.125 + 0.216i)16-s + (0.393 − 1.46i)17-s + (−1.18 − 1.18i)18-s + (0.783 − 0.620i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.426 + 0.904i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.426 + 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $0.426 + 0.904i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ 0.426 + 0.904i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.62540 - 1.66404i\)
\(L(\frac12)\) \(\approx\) \(2.62540 - 1.66404i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.965 - 0.258i)T \)
5 \( 1 \)
19 \( 1 + (-3.41 + 2.70i)T \)
good3 \( 1 + (-0.824 + 3.07i)T + (-2.59 - 1.5i)T^{2} \)
7 \( 1 + (-3.29 - 3.29i)T + 7iT^{2} \)
11 \( 1 - 1.07T + 11T^{2} \)
13 \( 1 + (0.991 - 0.265i)T + (11.2 - 6.5i)T^{2} \)
17 \( 1 + (-1.62 + 6.05i)T + (-14.7 - 8.5i)T^{2} \)
23 \( 1 + (-0.642 - 2.39i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + (-2.98 + 5.17i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 0.124iT - 31T^{2} \)
37 \( 1 + (3.61 - 3.61i)T - 37iT^{2} \)
41 \( 1 + (4.79 - 2.76i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (10.2 + 2.75i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 + (9.50 - 2.54i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (-1.72 + 0.463i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (-5.41 - 9.38i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.14 - 5.44i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.60 - 13.4i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (-9.49 + 5.48i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (-9.56 - 2.56i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (3.35 + 5.80i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (3.38 - 3.38i)T - 83iT^{2} \)
89 \( 1 + (-2.05 + 3.56i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (9.27 + 2.48i)T + (84.0 + 48.5i)T^{2} \)
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

−9.624722993203191453784746158890, −8.644565320127126067403691344450, −8.081134977874429604243761043638, −7.28403771945004831870528065293, −6.62088643116471345658089171297, −5.54133676172575602319436574964, −4.93516902109160358791486431162, −3.08995173315223762013764871273, −2.35911442551642042247717577339, −1.32962773359007769341670923804, 1.72736743477509224286408746925, 3.45552689285195960553125128961, 3.81321732527030759134360550956, 4.85974937370669035435598907111, 5.22521184419626757217090436068, 6.66890037428762944929149336809, 7.980904880481775593129119570283, 8.439232447032707974126375737433, 9.714766055303989839060063616518, 10.30987114358768312486177118072

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