Properties

Label 2-950-95.9-c1-0-9
Degree $2$
Conductor $950$
Sign $0.791 + 0.611i$
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  + (−0.984 + 0.173i)2-s + (−0.712 − 0.849i)3-s + (0.939 − 0.342i)4-s + (0.849 + 0.712i)6-s + (4.26 + 2.46i)7-s + (−0.866 + 0.5i)8-s + (0.307 − 1.74i)9-s + (−2.20 − 3.82i)11-s + (−0.960 − 0.554i)12-s + (−1.69 + 2.02i)13-s + (−4.63 − 1.68i)14-s + (0.766 − 0.642i)16-s + (4.94 − 0.872i)17-s + 1.76i·18-s + (4.21 − 1.11i)19-s + ⋯
L(s)  = 1  + (−0.696 + 0.122i)2-s + (−0.411 − 0.490i)3-s + (0.469 − 0.171i)4-s + (0.346 + 0.291i)6-s + (1.61 + 0.931i)7-s + (−0.306 + 0.176i)8-s + (0.102 − 0.580i)9-s + (−0.665 − 1.15i)11-s + (−0.277 − 0.160i)12-s + (−0.470 + 0.560i)13-s + (−1.23 − 0.450i)14-s + (0.191 − 0.160i)16-s + (1.20 − 0.211i)17-s + 0.417i·18-s + (0.966 − 0.256i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.791 + 0.611i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.791 + 0.611i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $0.791 + 0.611i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ 0.791 + 0.611i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.11096 - 0.379147i\)
\(L(\frac12)\) \(\approx\) \(1.11096 - 0.379147i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.984 - 0.173i)T \)
5 \( 1 \)
19 \( 1 + (-4.21 + 1.11i)T \)
good3 \( 1 + (0.712 + 0.849i)T + (-0.520 + 2.95i)T^{2} \)
7 \( 1 + (-4.26 - 2.46i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.20 + 3.82i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.69 - 2.02i)T + (-2.25 - 12.8i)T^{2} \)
17 \( 1 + (-4.94 + 0.872i)T + (15.9 - 5.81i)T^{2} \)
23 \( 1 + (-0.351 - 0.964i)T + (-17.6 + 14.7i)T^{2} \)
29 \( 1 + (0.462 - 2.62i)T + (-27.2 - 9.91i)T^{2} \)
31 \( 1 + (1.01 - 1.76i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 2.00iT - 37T^{2} \)
41 \( 1 + (-1.65 + 1.38i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 + (-0.516 + 1.41i)T + (-32.9 - 27.6i)T^{2} \)
47 \( 1 + (1.76 + 0.310i)T + (44.1 + 16.0i)T^{2} \)
53 \( 1 + (1.92 + 5.28i)T + (-40.6 + 34.0i)T^{2} \)
59 \( 1 + (2.44 + 13.8i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (-13.6 + 4.96i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (-13.4 - 2.36i)T + (62.9 + 22.9i)T^{2} \)
71 \( 1 + (-14.6 - 5.33i)T + (54.3 + 45.6i)T^{2} \)
73 \( 1 + (8.10 + 9.66i)T + (-12.6 + 71.8i)T^{2} \)
79 \( 1 + (-2.17 + 1.82i)T + (13.7 - 77.7i)T^{2} \)
83 \( 1 + (-6.30 - 3.64i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (7.72 + 6.48i)T + (15.4 + 87.6i)T^{2} \)
97 \( 1 + (-11.7 + 2.06i)T + (91.1 - 33.1i)T^{2} \)
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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

−9.796118484227086621993821370816, −9.036141742086372609828607874797, −8.187528394948958245341328904589, −7.63548677837546776008350178304, −6.66051733034021837449719045260, −5.47634274273515558733580344936, −5.21872284183770661770998790785, −3.36831639118600795067844949970, −2.04770627605178563732462659317, −0.902660616193618477344588321350, 1.18517585017799962834917494592, 2.38039357331253427074514638686, 4.00843573150368488079316730307, 4.95123381981736766559576137290, 5.50058412480142953627676189182, 7.19707115957701540812639191403, 7.77556091405842015707257410228, 8.121585880628497635832922451337, 9.678325905802976844056224358985, 10.16701954909302295929944454503

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