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Properties

Label 2-950-95.24-c1-0-13
Degree $2$
Conductor $950$
Sign $0.937 + 0.348i$
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 + (−1.85 − 0.326i)3-s + (−0.766 + 0.642i)4-s + (−0.326 − 1.85i)6-s + (−4.38 + 2.53i)7-s + (−0.866 − 0.500i)8-s + (0.5 + 0.181i)9-s + (0.705 − 1.22i)11-s + (1.62 − 0.939i)12-s + (1.28 − 0.226i)13-s + (−3.87 − 3.25i)14-s + (0.173 − 0.984i)16-s + (0.817 + 2.24i)17-s + 0.532i·18-s + (−2.23 − 3.74i)19-s + ⋯
L(s)  = 1  + (0.241 + 0.664i)2-s + (−1.06 − 0.188i)3-s + (−0.383 + 0.321i)4-s + (−0.133 − 0.755i)6-s + (−1.65 + 0.957i)7-s + (−0.306 − 0.176i)8-s + (0.166 + 0.0606i)9-s + (0.212 − 0.368i)11-s + (0.469 − 0.271i)12-s + (0.356 − 0.0628i)13-s + (−1.03 − 0.869i)14-s + (0.0434 − 0.246i)16-s + (0.198 + 0.544i)17-s + 0.125i·18-s + (−0.512 − 0.858i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.937 + 0.348i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.937 + 0.348i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $0.937 + 0.348i$
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: \((2,\ 950,\ (\ :1/2),\ 0.937 + 0.348i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.578952 - 0.104019i\)
\(L(\frac12)\) \(\approx\) \(0.578952 - 0.104019i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.23 + 3.74i)T \)
good3 \( 1 + (1.85 + 0.326i)T + (2.81 + 1.02i)T^{2} \)
7 \( 1 + (4.38 - 2.53i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.705 + 1.22i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.28 + 0.226i)T + (12.2 - 4.44i)T^{2} \)
17 \( 1 + (-0.817 - 2.24i)T + (-13.0 + 10.9i)T^{2} \)
23 \( 1 + (1.96 + 2.34i)T + (-3.99 + 22.6i)T^{2} \)
29 \( 1 + (-7.94 - 2.89i)T + (22.2 + 18.6i)T^{2} \)
31 \( 1 + (0.184 + 0.320i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 4.82iT - 37T^{2} \)
41 \( 1 + (0.266 - 1.50i)T + (-38.5 - 14.0i)T^{2} \)
43 \( 1 + (-0.487 + 0.581i)T + (-7.46 - 42.3i)T^{2} \)
47 \( 1 + (-3.49 + 9.59i)T + (-36.0 - 30.2i)T^{2} \)
53 \( 1 + (-1.07 - 1.28i)T + (-9.20 + 52.1i)T^{2} \)
59 \( 1 + (-0.673 + 0.245i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (-7.47 + 6.27i)T + (10.5 - 60.0i)T^{2} \)
67 \( 1 + (-0.480 + 1.31i)T + (-51.3 - 43.0i)T^{2} \)
71 \( 1 + (4.87 + 4.09i)T + (12.3 + 69.9i)T^{2} \)
73 \( 1 + (4.49 + 0.791i)T + (68.5 + 24.9i)T^{2} \)
79 \( 1 + (-0.389 + 2.20i)T + (-74.2 - 27.0i)T^{2} \)
83 \( 1 + (3.45 - 1.99i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (-1.84 - 10.4i)T + (-83.6 + 30.4i)T^{2} \)
97 \( 1 + (0.524 + 1.43i)T + (-74.3 + 62.3i)T^{2} \)
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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

−10.00156750394125634914287835967, −8.947858190196253481885728463952, −8.491000799088417043977935684070, −6.96666981761732119298205009847, −6.38436472179925167562209476477, −5.92906930949399490855618031894, −5.08905621116430002510160305562, −3.75509714571436578679510869356, −2.70524077417677992030758897301, −0.38455086387480233094104687602, 0.940264529552152716824966738260, 2.81661053996232694585438314755, 3.84329102246804096225703353369, 4.63196294656220123810520407576, 5.89616472743393590564025938147, 6.36214762722168009800353681529, 7.31698741409571861777916209448, 8.626750449295277546620420937910, 9.841832288684771874340596172169, 10.07417209917912587263967946378

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