Properties

Label 2-950-95.54-c1-0-11
Degree $2$
Conductor $950$
Sign $0.884 - 0.467i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.642 + 0.766i)2-s + (−0.0291 − 0.0801i)3-s + (−0.173 − 0.984i)4-s + (0.0801 + 0.0291i)6-s + (1.59 − 0.920i)7-s + (0.866 + 0.500i)8-s + (2.29 − 1.92i)9-s + (−1.21 + 2.09i)11-s + (−0.0738 + 0.0426i)12-s + (−1.03 + 2.84i)13-s + (−0.319 + 1.81i)14-s + (−0.939 + 0.342i)16-s + (−4.38 + 5.22i)17-s + 2.99i·18-s + (3.15 − 3.00i)19-s + ⋯
L(s)  = 1  + (−0.454 + 0.541i)2-s + (−0.0168 − 0.0462i)3-s + (−0.0868 − 0.492i)4-s + (0.0327 + 0.0119i)6-s + (0.602 − 0.347i)7-s + (0.306 + 0.176i)8-s + (0.764 − 0.641i)9-s + (−0.364 + 0.632i)11-s + (−0.0213 + 0.0123i)12-s + (−0.287 + 0.790i)13-s + (−0.0854 + 0.484i)14-s + (−0.234 + 0.0855i)16-s + (−1.06 + 1.26i)17-s + 0.705i·18-s + (0.724 − 0.689i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.884 - 0.467i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.884 - 0.467i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $0.884 - 0.467i$
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: \((2,\ 950,\ (\ :1/2),\ 0.884 - 0.467i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.33183 + 0.330351i\)
\(L(\frac12)\) \(\approx\) \(1.33183 + 0.330351i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.15 + 3.00i)T \)
good3 \( 1 + (0.0291 + 0.0801i)T + (-2.29 + 1.92i)T^{2} \)
7 \( 1 + (-1.59 + 0.920i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.21 - 2.09i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.03 - 2.84i)T + (-9.95 - 8.35i)T^{2} \)
17 \( 1 + (4.38 - 5.22i)T + (-2.95 - 16.7i)T^{2} \)
23 \( 1 + (-7.90 + 1.39i)T + (21.6 - 7.86i)T^{2} \)
29 \( 1 + (-7.77 + 6.52i)T + (5.03 - 28.5i)T^{2} \)
31 \( 1 + (2.41 + 4.18i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 0.202iT - 37T^{2} \)
41 \( 1 + (-5.41 + 1.97i)T + (31.4 - 26.3i)T^{2} \)
43 \( 1 + (2.07 + 0.365i)T + (40.4 + 14.7i)T^{2} \)
47 \( 1 + (-6.32 - 7.53i)T + (-8.16 + 46.2i)T^{2} \)
53 \( 1 + (-6.17 + 1.08i)T + (49.8 - 18.1i)T^{2} \)
59 \( 1 + (-4.62 - 3.87i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (-0.741 - 4.20i)T + (-57.3 + 20.8i)T^{2} \)
67 \( 1 + (-2.00 - 2.39i)T + (-11.6 + 65.9i)T^{2} \)
71 \( 1 + (1.94 - 11.0i)T + (-66.7 - 24.2i)T^{2} \)
73 \( 1 + (2.67 + 7.33i)T + (-55.9 + 46.9i)T^{2} \)
79 \( 1 + (-9.87 + 3.59i)T + (60.5 - 50.7i)T^{2} \)
83 \( 1 + (11.0 - 6.37i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (-1.21 - 0.442i)T + (68.1 + 57.2i)T^{2} \)
97 \( 1 + (-5.27 + 6.28i)T + (-16.8 - 95.5i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.962877203667178547063984252014, −9.214409050619715364318458438188, −8.472369791502356836798215822367, −7.37631921666845490240583268137, −6.95991730146945995155219422065, −6.01350812958269469006284784273, −4.67827584030352339785573106554, −4.22168316061812067554684065253, −2.38581813184700191623901604711, −1.06544535881955043974961913804, 1.03758381990098486579377669943, 2.41691180163736607757974065344, 3.34675825086080150698795693228, 4.87412723004641834379654663401, 5.24062865477113860107199864479, 6.88055810535836506742783956844, 7.54894745106416060946276179558, 8.458307753221126703661942094851, 9.110388646835439865167646887112, 10.12221022002966363184798620827

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