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Properties

Label 2-950-95.54-c1-0-23
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: $\chi_{950} (149, \cdot )$
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

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

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