Properties

Label 2-950-19.11-c1-0-31
Degree $2$
Conductor $950$
Sign $-0.988 - 0.149i$
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 + (1.63 − 2.82i)3-s + (−0.499 − 0.866i)4-s + (−1.63 − 2.82i)6-s − 2.62·7-s − 0.999·8-s + (−3.82 − 6.63i)9-s + 5.03·11-s − 3.26·12-s + (−2.32 − 4.02i)13-s + (−1.31 + 2.26i)14-s + (−0.5 + 0.866i)16-s + (1.82 − 3.16i)17-s − 7.65·18-s + (0.697 + 4.30i)19-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (0.942 − 1.63i)3-s + (−0.249 − 0.433i)4-s + (−0.666 − 1.15i)6-s − 0.990·7-s − 0.353·8-s + (−1.27 − 2.21i)9-s + 1.51·11-s − 0.942·12-s + (−0.644 − 1.11i)13-s + (−0.350 + 0.606i)14-s + (−0.125 + 0.216i)16-s + (0.443 − 0.768i)17-s − 1.80·18-s + (0.160 + 0.987i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.988 - 0.149i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.988 - 0.149i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $-0.988 - 0.149i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (201, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ -0.988 - 0.149i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.162285 + 2.15921i\)
\(L(\frac12)\) \(\approx\) \(0.162285 + 2.15921i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.697 - 4.30i)T \)
good3 \( 1 + (-1.63 + 2.82i)T + (-1.5 - 2.59i)T^{2} \)
7 \( 1 + 2.62T + 7T^{2} \)
11 \( 1 - 5.03T + 11T^{2} \)
13 \( 1 + (2.32 + 4.02i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-1.82 + 3.16i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (-2.34 - 4.05i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-4.01 - 6.96i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 3.28T + 31T^{2} \)
37 \( 1 + 5.75T + 37T^{2} \)
41 \( 1 + (-2.90 + 5.03i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.290 + 0.502i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (2.31 + 4.00i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2.31 - 4.00i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-1.88 + 3.26i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.0650 + 0.112i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.189 + 0.328i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-4.56 + 7.91i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-4.25 + 7.36i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-3.98 + 6.89i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 7.86T + 83T^{2} \)
89 \( 1 + (-4.14 - 7.17i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-8.09 + 14.0i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.373290546395516746670383378150, −8.950348926277662750506995031996, −7.80203398021737203278204091601, −7.05716922564123035576353957564, −6.36684780161480538335389115677, −5.37839904822000438447830701472, −3.44746303090690162788368293314, −3.24495861690733582389711505639, −1.90192225884330251792002311385, −0.822720642827889143756087787155, 2.52065732923529231598917618918, 3.57042328987009103114924013227, 4.18980146052272614897284223062, 4.93793048956109468808209022005, 6.23090524335363928401229545977, 6.95991361553962555434086669391, 8.224091973850637508967966255953, 9.028732904220166536308206384041, 9.453085405029618606649815325062, 10.07630049081937743630092950943

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