Properties

Label 2-950-19.7-c1-0-31
Degree $2$
Conductor $950$
Sign $-0.813 - 0.582i$
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  + (−0.5 − 0.866i)2-s + (−0.5 − 0.866i)3-s + (−0.499 + 0.866i)4-s + (−0.499 + 0.866i)6-s − 2·7-s + 0.999·8-s + (1 − 1.73i)9-s − 3·11-s + 0.999·12-s + (3 − 5.19i)13-s + (1 + 1.73i)14-s + (−0.5 − 0.866i)16-s + (1 + 1.73i)17-s − 2·18-s + (3.5 − 2.59i)19-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.288 − 0.499i)3-s + (−0.249 + 0.433i)4-s + (−0.204 + 0.353i)6-s − 0.755·7-s + 0.353·8-s + (0.333 − 0.577i)9-s − 0.904·11-s + 0.288·12-s + (0.832 − 1.44i)13-s + (0.267 + 0.462i)14-s + (−0.125 − 0.216i)16-s + (0.242 + 0.420i)17-s − 0.471·18-s + (0.802 − 0.596i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.813 - 0.582i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.813 - 0.582i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $-0.813 - 0.582i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (501, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ -0.813 - 0.582i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.122636 + 0.381867i\)
\(L(\frac12)\) \(\approx\) \(0.122636 + 0.381867i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 + 0.866i)T \)
5 \( 1 \)
19 \( 1 + (-3.5 + 2.59i)T \)
good3 \( 1 + (0.5 + 0.866i)T + (-1.5 + 2.59i)T^{2} \)
7 \( 1 + 2T + 7T^{2} \)
11 \( 1 + 3T + 11T^{2} \)
13 \( 1 + (-3 + 5.19i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-1 - 1.73i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (4 - 6.92i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1 + 1.73i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 8T + 31T^{2} \)
37 \( 1 + 8T + 37T^{2} \)
41 \( 1 + (2.5 + 4.33i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (3 - 5.19i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-3 + 5.19i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.5 + 4.33i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (7 - 12.1i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.5 - 4.33i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-3 - 5.19i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (4.5 + 7.79i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (4 + 6.92i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 11T + 83T^{2} \)
89 \( 1 + (7 - 12.1i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (7.5 + 12.9i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.733230303290900359360727722307, −8.809603457363235577501433377780, −7.80166390321345304139152639750, −7.20960670757774901493112066904, −6.00537553320877504752555515351, −5.35379623129920964278335808955, −3.68746108087346518524154159102, −3.11755097200733550119288067858, −1.55806517875272411449924869470, −0.21899355817458887370701028620, 1.86194946388814138926680489730, 3.47455028564427821478408444633, 4.54638968321674808446313788390, 5.39971952835208150949969959237, 6.31965801127772253482119207790, 7.12708564082600825366052882373, 8.016702852100782720064155637798, 8.925555673642421527702407336663, 9.723062748053496413028711491741, 10.36859661640223438742265491932

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