Properties

Label 2-950-1.1-c1-0-21
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 + 2.25·3-s + 4-s + 2.25·6-s + 4.22·7-s + 8-s + 2.08·9-s − 5.13·11-s + 2.25·12-s + 3.16·13-s + 4.22·14-s + 16-s − 6.48·17-s + 2.08·18-s − 19-s + 9.53·21-s − 5.13·22-s + 7.56·23-s + 2.25·24-s + 3.16·26-s − 2.05·27-s + 4.22·28-s + 0.832·29-s − 4.51·31-s + 32-s − 11.5·33-s − 6.48·34-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.30·3-s + 0.5·4-s + 0.920·6-s + 1.59·7-s + 0.353·8-s + 0.695·9-s − 1.54·11-s + 0.651·12-s + 0.878·13-s + 1.12·14-s + 0.250·16-s − 1.57·17-s + 0.492·18-s − 0.229·19-s + 2.07·21-s − 1.09·22-s + 1.57·23-s + 0.460·24-s + 0.621·26-s − 0.395·27-s + 0.798·28-s + 0.154·29-s − 0.810·31-s + 0.176·32-s − 2.01·33-s − 1.11·34-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

\(L(1)\) \(\approx\) \(3.969657378\)
\(L(\frac12)\) \(\approx\) \(3.969657378\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.25T + 3T^{2} \)
7 \( 1 - 4.22T + 7T^{2} \)
11 \( 1 + 5.13T + 11T^{2} \)
13 \( 1 - 3.16T + 13T^{2} \)
17 \( 1 + 6.48T + 17T^{2} \)
23 \( 1 - 7.56T + 23T^{2} \)
29 \( 1 - 0.832T + 29T^{2} \)
31 \( 1 + 4.51T + 31T^{2} \)
37 \( 1 - 0.137T + 37T^{2} \)
41 \( 1 + 11.6T + 41T^{2} \)
43 \( 1 + 2.51T + 43T^{2} \)
47 \( 1 + 5.96T + 47T^{2} \)
53 \( 1 - 0.225T + 53T^{2} \)
59 \( 1 - 5.39T + 59T^{2} \)
61 \( 1 - 14.4T + 61T^{2} \)
67 \( 1 + 4.11T + 67T^{2} \)
71 \( 1 - 3.82T + 71T^{2} \)
73 \( 1 - 4.70T + 73T^{2} \)
79 \( 1 - 10.6T + 79T^{2} \)
83 \( 1 + 12.0T + 83T^{2} \)
89 \( 1 + 10T + 89T^{2} \)
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

−10.15404365055772919853424609626, −8.712602781636527498136243445687, −8.496889024136592306280115373498, −7.65862334787671301641646781821, −6.78734593420353788327942476600, −5.33188401267888014535227763496, −4.76445659554234147987757244775, −3.65140799252629995017406128079, −2.59022385295220915040614234210, −1.79621579347191580862917911994, 1.79621579347191580862917911994, 2.59022385295220915040614234210, 3.65140799252629995017406128079, 4.76445659554234147987757244775, 5.33188401267888014535227763496, 6.78734593420353788327942476600, 7.65862334787671301641646781821, 8.496889024136592306280115373498, 8.712602781636527498136243445687, 10.15404365055772919853424609626

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