Properties

Label 2-950-19.11-c1-0-25
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 + (−1.39 + 2.41i)3-s + (−0.499 − 0.866i)4-s + (1.39 + 2.41i)6-s + 7-s − 0.999·8-s + (−2.39 − 4.14i)9-s − 3.79·11-s + 2.79·12-s + (−0.104 − 0.180i)13-s + (0.5 − 0.866i)14-s + (−0.5 + 0.866i)16-s + (−0.395 + 0.685i)17-s − 4.79·18-s + (−3.5 − 2.59i)19-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.805 + 1.39i)3-s + (−0.249 − 0.433i)4-s + (0.569 + 0.986i)6-s + 0.377·7-s − 0.353·8-s + (−0.798 − 1.38i)9-s − 1.14·11-s + 0.805·12-s + (−0.0289 − 0.0501i)13-s + (0.133 − 0.231i)14-s + (−0.125 + 0.216i)16-s + (−0.0959 + 0.166i)17-s − 1.12·18-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: $\chi_{950} (201, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ -0.813 + 0.582i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0708259 - 0.220538i\)
\(L(\frac12)\) \(\approx\) \(0.0708259 - 0.220538i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.39 - 2.41i)T + (-1.5 - 2.59i)T^{2} \)
7 \( 1 - T + 7T^{2} \)
11 \( 1 + 3.79T + 11T^{2} \)
13 \( 1 + (0.104 + 0.180i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (0.395 - 0.685i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (-2.29 - 3.96i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.39 + 5.88i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 4.79T + 31T^{2} \)
37 \( 1 + 3.58T + 37T^{2} \)
41 \( 1 + (-5.68 + 9.85i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.89 + 5.01i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (6.08 + 10.5i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (2.29 + 3.96i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (2.29 - 3.96i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.686 + 1.18i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-7.79 - 13.4i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-2.29 + 3.96i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-1.39 + 2.41i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (7.47 - 12.9i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 3.79T + 83T^{2} \)
89 \( 1 + (2.29 + 3.96i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (6.18 - 10.7i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.949154829370621062128039307794, −9.240458781463918799138786150700, −8.279589469449587235389293294450, −7.01675482629958724454542097448, −5.61871130393530834677856707584, −5.30827631537671279337975948631, −4.34493216790591419390008030818, −3.57416520509140839848254950579, −2.23301595935747769826021735541, −0.10217185655563589183540727933, 1.61243259455048431009984848406, 2.89791544025812513703614700364, 4.57652121498817056118746524337, 5.37474007567763474102953803209, 6.19375803704122686745697877803, 6.87732663761693864358848991356, 7.78933389533083435830593608459, 8.151757904243355665385491907721, 9.378437223703305077800482018453, 10.79768733659773530509477272183

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