Properties

Label 2-950-95.44-c1-0-9
Degree $2$
Conductor $950$
Sign $0.648 - 0.761i$
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 + (−0.197 + 0.541i)3-s + (−0.173 + 0.984i)4-s + (0.541 − 0.197i)6-s + (4.21 + 2.43i)7-s + (0.866 − 0.500i)8-s + (2.04 + 1.71i)9-s + (2.68 + 4.64i)11-s + (−0.499 − 0.288i)12-s + (−1.31 − 3.62i)13-s + (−0.844 − 4.79i)14-s + (−0.939 − 0.342i)16-s + (0.901 + 1.07i)17-s − 2.66i·18-s + (−4.35 − 0.226i)19-s + ⋯
L(s)  = 1  + (−0.454 − 0.541i)2-s + (−0.113 + 0.312i)3-s + (−0.0868 + 0.492i)4-s + (0.221 − 0.0804i)6-s + (1.59 + 0.919i)7-s + (0.306 − 0.176i)8-s + (0.681 + 0.571i)9-s + (0.809 + 1.40i)11-s + (−0.144 − 0.0831i)12-s + (−0.365 − 1.00i)13-s + (−0.225 − 1.28i)14-s + (−0.234 − 0.0855i)16-s + (0.218 + 0.260i)17-s − 0.628i·18-s + (−0.998 − 0.0520i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.648 - 0.761i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.648 - 0.761i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $0.648 - 0.761i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (899, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ 0.648 - 0.761i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.31657 + 0.608102i\)
\(L(\frac12)\) \(\approx\) \(1.31657 + 0.608102i\)
\(L(\frac{3}{2})\) 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 \( 1 + (4.35 + 0.226i)T \)
good3 \( 1 + (0.197 - 0.541i)T + (-2.29 - 1.92i)T^{2} \)
7 \( 1 + (-4.21 - 2.43i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.68 - 4.64i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.31 + 3.62i)T + (-9.95 + 8.35i)T^{2} \)
17 \( 1 + (-0.901 - 1.07i)T + (-2.95 + 16.7i)T^{2} \)
23 \( 1 + (5.25 + 0.927i)T + (21.6 + 7.86i)T^{2} \)
29 \( 1 + (-2.78 - 2.33i)T + (5.03 + 28.5i)T^{2} \)
31 \( 1 + (4.10 - 7.10i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 10.4iT - 37T^{2} \)
41 \( 1 + (-1.79 - 0.652i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (-1.45 + 0.256i)T + (40.4 - 14.7i)T^{2} \)
47 \( 1 + (-2.00 + 2.39i)T + (-8.16 - 46.2i)T^{2} \)
53 \( 1 + (-1.77 - 0.312i)T + (49.8 + 18.1i)T^{2} \)
59 \( 1 + (-5.61 + 4.70i)T + (10.2 - 58.1i)T^{2} \)
61 \( 1 + (-1.40 + 7.99i)T + (-57.3 - 20.8i)T^{2} \)
67 \( 1 + (5.42 - 6.46i)T + (-11.6 - 65.9i)T^{2} \)
71 \( 1 + (-1.04 - 5.93i)T + (-66.7 + 24.2i)T^{2} \)
73 \( 1 + (-0.436 + 1.19i)T + (-55.9 - 46.9i)T^{2} \)
79 \( 1 + (15.6 + 5.68i)T + (60.5 + 50.7i)T^{2} \)
83 \( 1 + (-12.5 - 7.23i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (-9.02 + 3.28i)T + (68.1 - 57.2i)T^{2} \)
97 \( 1 + (3.90 + 4.65i)T + (-16.8 + 95.5i)T^{2} \)
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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

−10.34201100566909942098805555302, −9.374305321370710886437223388816, −8.544305756640830270426057295930, −7.83009591228322526030853210392, −7.05329435954354189040950793604, −5.56262101714495630135723422977, −4.75052663883449030402832874083, −4.02119228849092569273579050530, −2.26397547691918674342033452399, −1.66334261259586999762366770827, 0.879662470733886796715314649133, 1.86275897882785912582761617588, 3.96981138012998157226520049843, 4.50298257802120206214618030319, 5.87194072928934774489098524334, 6.61157177422547487166872412638, 7.46541618523230704114894755121, 8.153419510913036032153227292836, 8.917644398569347590534783580784, 9.856947618208440610245157228501

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