Properties

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

\(L(1)\) \(\approx\) \(0.2312192921\)
\(L(\frac12)\) \(\approx\) \(0.2312192921\)
\(L(\frac{3}{2})\) 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

−10.16863290573775817722242104543, −9.614188778614682701766833594526, −8.282540130943472470637899407321, −7.42610218019810107613377226861, −6.62229112629908940967412365005, −5.73763677719512194073549486951, −5.22847686210246455263254895493, −3.57201579756469169734144714353, −2.44082091680374607224470303204, −0.41536246201856860465777332556, 0.41536246201856860465777332556, 2.44082091680374607224470303204, 3.57201579756469169734144714353, 5.22847686210246455263254895493, 5.73763677719512194073549486951, 6.62229112629908940967412365005, 7.42610218019810107613377226861, 8.282540130943472470637899407321, 9.614188778614682701766833594526, 10.16863290573775817722242104543

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