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Properties

Label 2-950-95.24-c1-0-6
Degree $2$
Conductor $950$
Sign $-0.113 - 0.993i$
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  + (0.342 + 0.939i)2-s + (−2.49 − 0.439i)3-s + (−0.766 + 0.642i)4-s + (−0.439 − 2.49i)6-s + (0.565 − 0.326i)7-s + (−0.866 − 0.500i)8-s + (3.20 + 1.16i)9-s + (0.5 − 0.866i)11-s + (2.19 − 1.26i)12-s + (−2.83 + 0.5i)13-s + (0.5 + 0.419i)14-s + (0.173 − 0.984i)16-s + (0.160 + 0.439i)17-s + 3.41i·18-s + (4.07 − 1.55i)19-s + ⋯
L(s)  = 1  + (0.241 + 0.664i)2-s + (−1.43 − 0.253i)3-s + (−0.383 + 0.321i)4-s + (−0.179 − 1.01i)6-s + (0.213 − 0.123i)7-s + (−0.306 − 0.176i)8-s + (1.06 + 0.388i)9-s + (0.150 − 0.261i)11-s + (0.633 − 0.365i)12-s + (−0.786 + 0.138i)13-s + (0.133 + 0.112i)14-s + (0.0434 − 0.246i)16-s + (0.0388 + 0.106i)17-s + 0.804i·18-s + (0.934 − 0.355i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.113 - 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.113 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $-0.113 - 0.993i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (499, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ -0.113 - 0.993i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.556124 + 0.623071i\)
\(L(\frac12)\) \(\approx\) \(0.556124 + 0.623071i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.342 - 0.939i)T \)
5 \( 1 \)
19 \( 1 + (-4.07 + 1.55i)T \)
good3 \( 1 + (2.49 + 0.439i)T + (2.81 + 1.02i)T^{2} \)
7 \( 1 + (-0.565 + 0.326i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.5 + 0.866i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.83 - 0.5i)T + (12.2 - 4.44i)T^{2} \)
17 \( 1 + (-0.160 - 0.439i)T + (-13.0 + 10.9i)T^{2} \)
23 \( 1 + (2.15 + 2.56i)T + (-3.99 + 22.6i)T^{2} \)
29 \( 1 + (-6.41 - 2.33i)T + (22.2 + 18.6i)T^{2} \)
31 \( 1 + (-2.03 - 3.51i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 7.63iT - 37T^{2} \)
41 \( 1 + (0.854 - 4.84i)T + (-38.5 - 14.0i)T^{2} \)
43 \( 1 + (2.02 - 2.40i)T + (-7.46 - 42.3i)T^{2} \)
47 \( 1 + (2.19 - 6.02i)T + (-36.0 - 30.2i)T^{2} \)
53 \( 1 + (-2.48 - 2.96i)T + (-9.20 + 52.1i)T^{2} \)
59 \( 1 + (5.68 - 2.06i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (3.96 - 3.32i)T + (10.5 - 60.0i)T^{2} \)
67 \( 1 + (-0.300 + 0.826i)T + (-51.3 - 43.0i)T^{2} \)
71 \( 1 + (-9.35 - 7.84i)T + (12.3 + 69.9i)T^{2} \)
73 \( 1 + (-0.502 - 0.0885i)T + (68.5 + 24.9i)T^{2} \)
79 \( 1 + (-1.51 + 8.58i)T + (-74.2 - 27.0i)T^{2} \)
83 \( 1 + (-15.5 + 8.95i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (-3.06 - 17.3i)T + (-83.6 + 30.4i)T^{2} \)
97 \( 1 + (-4.96 - 13.6i)T + (-74.3 + 62.3i)T^{2} \)
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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

−10.36028241251770807839705404961, −9.506779589625309230600995914064, −8.367095225555529832939293983361, −7.49523057377564216751163676968, −6.61166297903577517063535622976, −6.11785466005769882262210207253, −4.98980087088827485687915923887, −4.64581053086734859835783866777, −3.03277515073074198342701845526, −1.07306357382088798748298503020, 0.54037892619563938913221904294, 2.09261084514826961801628982874, 3.57517737853925936382622558921, 4.69563491591073315482076604672, 5.28915687269237086077928210459, 6.08549294379754182075654541422, 7.09499127170396803503872140933, 8.141403042722843819208741868667, 9.440308617257373076388755752735, 10.03451166193428990743867036151

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