↑

Properties

Label 2-950-25.6-c1-0-8
Degree $2$
Conductor $950$
Sign $-0.728 - 0.685i$
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  + (0.309 + 0.951i)2-s + (−0.0867 + 0.0630i)3-s + (−0.809 + 0.587i)4-s + (2.07 − 0.821i)5-s + (−0.0867 − 0.0630i)6-s − 1.31·7-s + (−0.809 − 0.587i)8-s + (−0.923 + 2.84i)9-s + (1.42 + 1.72i)10-s + (0.239 + 0.736i)11-s + (0.0331 − 0.101i)12-s + (−1.58 + 4.87i)13-s + (−0.406 − 1.25i)14-s + (−0.128 + 0.202i)15-s + (0.309 − 0.951i)16-s + (−4.24 − 3.08i)17-s + ⋯
L(s)  = 1  + (0.218 + 0.672i)2-s + (−0.0500 + 0.0363i)3-s + (−0.404 + 0.293i)4-s + (0.930 − 0.367i)5-s + (−0.0354 − 0.0257i)6-s − 0.496·7-s + (−0.286 − 0.207i)8-s + (−0.307 + 0.947i)9-s + (0.450 + 0.545i)10-s + (0.0721 + 0.222i)11-s + (0.00956 − 0.0294i)12-s + (−0.438 + 1.35i)13-s + (−0.108 − 0.334i)14-s + (−0.0331 + 0.0522i)15-s + (0.0772 − 0.237i)16-s + (−1.03 − 0.748i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.728 - 0.685i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.728 - 0.685i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $-0.728 - 0.685i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (381, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ -0.728 - 0.685i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.521402 + 1.31429i\)
\(L(\frac12)\) \(\approx\) \(0.521402 + 1.31429i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.309 - 0.951i)T \)
5 \( 1 + (-2.07 + 0.821i)T \)
19 \( 1 + (0.809 + 0.587i)T \)
good3 \( 1 + (0.0867 - 0.0630i)T + (0.927 - 2.85i)T^{2} \)
7 \( 1 + 1.31T + 7T^{2} \)
11 \( 1 + (-0.239 - 0.736i)T + (-8.89 + 6.46i)T^{2} \)
13 \( 1 + (1.58 - 4.87i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (4.24 + 3.08i)T + (5.25 + 16.1i)T^{2} \)
23 \( 1 + (-2.83 - 8.71i)T + (-18.6 + 13.5i)T^{2} \)
29 \( 1 + (0.0651 - 0.0473i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-5.47 - 3.97i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (2.07 - 6.38i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (1.94 - 5.98i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 7.26T + 43T^{2} \)
47 \( 1 + (2.79 - 2.03i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (1.63 - 1.18i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-1.90 + 5.85i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-1.38 - 4.26i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + (6.44 + 4.68i)T + (20.7 + 63.7i)T^{2} \)
71 \( 1 + (-0.0703 + 0.0511i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (3.54 + 10.9i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-13.1 + 9.54i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (1.86 + 1.35i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + (1.34 + 4.12i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (0.825 - 0.599i)T + (29.9 - 92.2i)T^{2} \)
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

−10.09274666533852216799317227699, −9.345117708066528859225325621476, −8.863506196839934653349831587181, −7.70581724126960781560837130875, −6.80069806069562064263726711061, −6.17952942013151388219057757851, −4.98660311230839873998338190752, −4.64498495436631401718443028391, −2.99543251028558302850585889505, −1.80152935723309457848688091150, 0.59754126605846433586658874186, 2.32575054446978730447530577567, 3.06936956471522652182567807559, 4.19490138493830010287611657834, 5.46044201402346496543609567628, 6.19461825497612810250966236581, 6.86480821489886036357829924412, 8.358899840949697537499521707882, 9.056963919181239369436148781889, 9.906403301814717336946386829085

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