Properties

Label 2-950-1.1-c1-0-19
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 2.77·3-s + 4-s + 2.77·6-s − 4.69·7-s + 8-s + 4.71·9-s + 6.40·11-s + 2.77·12-s + 1.06·13-s − 4.69·14-s + 16-s + 1.91·17-s + 4.71·18-s − 19-s − 13.0·21-s + 6.40·22-s + 1.79·23-s + 2.77·24-s + 1.06·26-s + 4.75·27-s − 4.69·28-s + 2.93·29-s − 5.55·31-s + 32-s + 17.7·33-s + 1.91·34-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.60·3-s + 0.5·4-s + 1.13·6-s − 1.77·7-s + 0.353·8-s + 1.57·9-s + 1.93·11-s + 0.801·12-s + 0.295·13-s − 1.25·14-s + 0.250·16-s + 0.465·17-s + 1.11·18-s − 0.229·19-s − 2.84·21-s + 1.36·22-s + 0.374·23-s + 0.566·24-s + 0.208·26-s + 0.915·27-s − 0.887·28-s + 0.545·29-s − 0.997·31-s + 0.176·32-s + 3.09·33-s + 0.328·34-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

\(L(1)\) \(\approx\) \(3.842969743\)
\(L(\frac12)\) \(\approx\) \(3.842969743\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.77T + 3T^{2} \)
7 \( 1 + 4.69T + 7T^{2} \)
11 \( 1 - 6.40T + 11T^{2} \)
13 \( 1 - 1.06T + 13T^{2} \)
17 \( 1 - 1.91T + 17T^{2} \)
23 \( 1 - 1.79T + 23T^{2} \)
29 \( 1 - 2.93T + 29T^{2} \)
31 \( 1 + 5.55T + 31T^{2} \)
37 \( 1 + 11.4T + 37T^{2} \)
41 \( 1 + 1.14T + 41T^{2} \)
43 \( 1 + 3.55T + 43T^{2} \)
47 \( 1 - 10.8T + 47T^{2} \)
53 \( 1 + 8.69T + 53T^{2} \)
59 \( 1 + 5.63T + 59T^{2} \)
61 \( 1 + 3.39T + 61T^{2} \)
67 \( 1 + 8.82T + 67T^{2} \)
71 \( 1 + 1.42T + 71T^{2} \)
73 \( 1 + 12.6T + 73T^{2} \)
79 \( 1 + 1.96T + 79T^{2} \)
83 \( 1 - 16.2T + 83T^{2} \)
89 \( 1 + 10T + 89T^{2} \)
97 \( 1 + 14.9T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.762641523858283954421643493673, −9.163397354863053682121633926093, −8.626159814920393922514024077340, −7.28025840620011693446166694910, −6.73853777901954683676849324847, −5.89772105921229902687536822883, −4.23652763277160090553895259888, −3.49589122725442479360417953773, −3.05165975502807886761171049816, −1.63930587319736415565967290568, 1.63930587319736415565967290568, 3.05165975502807886761171049816, 3.49589122725442479360417953773, 4.23652763277160090553895259888, 5.89772105921229902687536822883, 6.73853777901954683676849324847, 7.28025840620011693446166694910, 8.626159814920393922514024077340, 9.163397354863053682121633926093, 9.762641523858283954421643493673

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