Properties

Label 2-950-1.1-c1-0-9
Degree $2$
Conductor $950$
Sign $1$
Analytic cond. $7.58578$
Root an. cond. $2.75423$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 2.12·3-s + 4-s − 2.12·6-s + 4.12·7-s + 8-s + 1.51·9-s − 2.64·11-s − 2.12·12-s + 2.51·13-s + 4.12·14-s + 16-s − 0.515·17-s + 1.51·18-s + 19-s − 8.76·21-s − 2.64·22-s + 3.09·23-s − 2.12·24-s + 2.51·26-s + 3.15·27-s + 4.12·28-s − 7.79·29-s + 3.67·31-s + 32-s + 5.60·33-s − 0.515·34-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.22·3-s + 0.5·4-s − 0.867·6-s + 1.55·7-s + 0.353·8-s + 0.505·9-s − 0.795·11-s − 0.613·12-s + 0.697·13-s + 1.10·14-s + 0.250·16-s − 0.124·17-s + 0.357·18-s + 0.229·19-s − 1.91·21-s − 0.562·22-s + 0.645·23-s − 0.433·24-s + 0.493·26-s + 0.607·27-s + 0.779·28-s − 1.44·29-s + 0.659·31-s + 0.176·32-s + 0.976·33-s − 0.0883·34-s + ⋯

Functional equation

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

Invariants

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

Particular Values

\(L(1)\) \(\approx\) \(1.904794654\)
\(L(\frac12)\) \(\approx\) \(1.904794654\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
5 \( 1 \)
19 \( 1 - T \)
good3 \( 1 + 2.12T + 3T^{2} \)
7 \( 1 - 4.12T + 7T^{2} \)
11 \( 1 + 2.64T + 11T^{2} \)
13 \( 1 - 2.51T + 13T^{2} \)
17 \( 1 + 0.515T + 17T^{2} \)
23 \( 1 - 3.09T + 23T^{2} \)
29 \( 1 + 7.79T + 29T^{2} \)
31 \( 1 - 3.67T + 31T^{2} \)
37 \( 1 - 10.2T + 37T^{2} \)
41 \( 1 - 8.88T + 41T^{2} \)
43 \( 1 - 8.64T + 43T^{2} \)
47 \( 1 + 4.96T + 47T^{2} \)
53 \( 1 - 5.48T + 53T^{2} \)
59 \( 1 - 3.15T + 59T^{2} \)
61 \( 1 + 12.6T + 61T^{2} \)
67 \( 1 + 7.40T + 67T^{2} \)
71 \( 1 - 11.1T + 71T^{2} \)
73 \( 1 + 2.70T + 73T^{2} \)
79 \( 1 - 16.7T + 79T^{2} \)
83 \( 1 - 3.28T + 83T^{2} \)
89 \( 1 + 7.60T + 89T^{2} \)
97 \( 1 - 3.93T + 97T^{2} \)
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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

−10.62504019786217100001680222205, −9.256936543336738859630471737407, −8.066584754341174477718301840495, −7.47863101195266106039433056515, −6.26737111533225019899374025043, −5.56426547275458976874290920043, −4.93123364380885842918339897804, −4.12689746501296188096492043279, −2.54368697698038401537038750792, −1.12642417778193919113120049963, 1.12642417778193919113120049963, 2.54368697698038401537038750792, 4.12689746501296188096492043279, 4.93123364380885842918339897804, 5.56426547275458976874290920043, 6.26737111533225019899374025043, 7.47863101195266106039433056515, 8.066584754341174477718301840495, 9.256936543336738859630471737407, 10.62504019786217100001680222205

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