↑

Properties

Label 2-950-95.54-c1-0-8
Degree $2$
Conductor $950$
Sign $-0.922 + 0.386i$
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  + (−0.642 + 0.766i)2-s + (1.01 + 2.78i)3-s + (−0.173 − 0.984i)4-s + (−2.78 − 1.01i)6-s + (0.787 − 0.454i)7-s + (0.866 + 0.500i)8-s + (−4.44 + 3.72i)9-s + (−1.82 + 3.16i)11-s + (2.56 − 1.48i)12-s + (−0.969 + 2.66i)13-s + (−0.157 + 0.895i)14-s + (−0.939 + 0.342i)16-s + (3.46 − 4.13i)17-s − 5.80i·18-s + (−3.41 + 2.70i)19-s + ⋯
L(s)  = 1  + (−0.454 + 0.541i)2-s + (0.585 + 1.60i)3-s + (−0.0868 − 0.492i)4-s + (−1.13 − 0.414i)6-s + (0.297 − 0.171i)7-s + (0.306 + 0.176i)8-s + (−1.48 + 1.24i)9-s + (−0.550 + 0.953i)11-s + (0.741 − 0.428i)12-s + (−0.268 + 0.738i)13-s + (−0.0422 + 0.239i)14-s + (−0.234 + 0.0855i)16-s + (0.840 − 1.00i)17-s − 1.36i·18-s + (−0.783 + 0.621i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.922 + 0.386i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.922 + 0.386i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $-0.922 + 0.386i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (149, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ -0.922 + 0.386i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.219911 - 1.09459i\)
\(L(\frac12)\) \(\approx\) \(0.219911 - 1.09459i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.642 - 0.766i)T \)
5 \( 1 \)
19 \( 1 + (3.41 - 2.70i)T \)
good3 \( 1 + (-1.01 - 2.78i)T + (-2.29 + 1.92i)T^{2} \)
7 \( 1 + (-0.787 + 0.454i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.82 - 3.16i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.969 - 2.66i)T + (-9.95 - 8.35i)T^{2} \)
17 \( 1 + (-3.46 + 4.13i)T + (-2.95 - 16.7i)T^{2} \)
23 \( 1 + (1.28 - 0.226i)T + (21.6 - 7.86i)T^{2} \)
29 \( 1 + (3.16 - 2.65i)T + (5.03 - 28.5i)T^{2} \)
31 \( 1 + (2.98 + 5.16i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 9.82iT - 37T^{2} \)
41 \( 1 + (0.0389 - 0.0141i)T + (31.4 - 26.3i)T^{2} \)
43 \( 1 + (-8.74 - 1.54i)T + (40.4 + 14.7i)T^{2} \)
47 \( 1 + (1.46 + 1.74i)T + (-8.16 + 46.2i)T^{2} \)
53 \( 1 + (10.0 - 1.76i)T + (49.8 - 18.1i)T^{2} \)
59 \( 1 + (4.30 + 3.61i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (0.646 + 3.66i)T + (-57.3 + 20.8i)T^{2} \)
67 \( 1 + (-8.80 - 10.4i)T + (-11.6 + 65.9i)T^{2} \)
71 \( 1 + (-1.01 + 5.75i)T + (-66.7 - 24.2i)T^{2} \)
73 \( 1 + (4.81 + 13.2i)T + (-55.9 + 46.9i)T^{2} \)
79 \( 1 + (-14.0 + 5.12i)T + (60.5 - 50.7i)T^{2} \)
83 \( 1 + (-4.15 + 2.39i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (-8.41 - 3.06i)T + (68.1 + 57.2i)T^{2} \)
97 \( 1 + (-10.7 + 12.7i)T + (-16.8 - 95.5i)T^{2} \)
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

−10.21534484670896432908975645063, −9.556017248561386335637168437142, −9.135170661588981313640523604209, −8.004679792772406187935473150875, −7.50026188949273623431415932972, −6.16441084298463571970025573779, −4.96260077644329667326045483143, −4.57408279628884182409356704081, −3.42492155093067180166454905191, −2.09922090625545544977701997164, 0.54400583318733284864668752194, 1.81922017145889590479350970718, 2.70292887780417690745210848251, 3.67313513235149559271160552177, 5.43769637750152884338518849494, 6.28657352728681439435196842435, 7.38045692215247511653116510716, 8.013131119632584329818987882079, 8.482635600283628270532462123292, 9.349165483812297202724953361884

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