Properties

Label 2-950-1.1-c1-0-8
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 − 1.56·3-s + 4-s − 1.56·6-s − 1.56·7-s + 8-s − 0.561·9-s + 4·11-s − 1.56·12-s + 6.68·13-s − 1.56·14-s + 16-s − 7.56·17-s − 0.561·18-s − 19-s + 2.43·21-s + 4·22-s + 4.68·23-s − 1.56·24-s + 6.68·26-s + 5.56·27-s − 1.56·28-s + 6.68·29-s + 3.12·31-s + 32-s − 6.24·33-s − 7.56·34-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.901·3-s + 0.5·4-s − 0.637·6-s − 0.590·7-s + 0.353·8-s − 0.187·9-s + 1.20·11-s − 0.450·12-s + 1.85·13-s − 0.417·14-s + 0.250·16-s − 1.83·17-s − 0.132·18-s − 0.229·19-s + 0.532·21-s + 0.852·22-s + 0.976·23-s − 0.318·24-s + 1.31·26-s + 1.07·27-s − 0.295·28-s + 1.24·29-s + 0.560·31-s + 0.176·32-s − 1.08·33-s − 1.29·34-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

\(L(1)\) \(\approx\) \(1.824081225\)
\(L(\frac12)\) \(\approx\) \(1.824081225\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.56T + 3T^{2} \)
7 \( 1 + 1.56T + 7T^{2} \)
11 \( 1 - 4T + 11T^{2} \)
13 \( 1 - 6.68T + 13T^{2} \)
17 \( 1 + 7.56T + 17T^{2} \)
23 \( 1 - 4.68T + 23T^{2} \)
29 \( 1 - 6.68T + 29T^{2} \)
31 \( 1 - 3.12T + 31T^{2} \)
37 \( 1 - 6T + 37T^{2} \)
41 \( 1 + 4.24T + 41T^{2} \)
43 \( 1 - 11.1T + 43T^{2} \)
47 \( 1 - 10.2T + 47T^{2} \)
53 \( 1 - 0.438T + 53T^{2} \)
59 \( 1 + 1.56T + 59T^{2} \)
61 \( 1 - 2.87T + 61T^{2} \)
67 \( 1 + 1.56T + 67T^{2} \)
71 \( 1 + 6.24T + 71T^{2} \)
73 \( 1 + 10.6T + 73T^{2} \)
79 \( 1 - 3.12T + 79T^{2} \)
83 \( 1 + 11.1T + 83T^{2} \)
89 \( 1 - 2T + 89T^{2} \)
97 \( 1 + 6T + 97T^{2} \)
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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

−10.40509723935053206255780021760, −9.030349635085689639443277468277, −8.607425023989674458271902987602, −7.00515204683514271804791357069, −6.25170554024009670678048785667, −6.06325101073125175662248675889, −4.67093250850662199388310576228, −3.94670328982411401578264404479, −2.76636362403408315057351541959, −1.06121728191310280520996406203, 1.06121728191310280520996406203, 2.76636362403408315057351541959, 3.94670328982411401578264404479, 4.67093250850662199388310576228, 6.06325101073125175662248675889, 6.25170554024009670678048785667, 7.00515204683514271804791357069, 8.607425023989674458271902987602, 9.030349635085689639443277468277, 10.40509723935053206255780021760

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