Properties

Label 2-950-1.1-c1-0-7
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 − 0.414·3-s + 4-s + 0.414·6-s + 4.41·7-s − 8-s − 2.82·9-s − 1.41·11-s − 0.414·12-s + 5.82·13-s − 4.41·14-s + 16-s + 17-s + 2.82·18-s − 19-s − 1.82·21-s + 1.41·22-s + 0.757·23-s + 0.414·24-s − 5.82·26-s + 2.41·27-s + 4.41·28-s − 0.171·29-s + 6.24·31-s − 32-s + 0.585·33-s − 34-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.239·3-s + 0.5·4-s + 0.169·6-s + 1.66·7-s − 0.353·8-s − 0.942·9-s − 0.426·11-s − 0.119·12-s + 1.61·13-s − 1.17·14-s + 0.250·16-s + 0.242·17-s + 0.666·18-s − 0.229·19-s − 0.398·21-s + 0.301·22-s + 0.157·23-s + 0.0845·24-s − 1.14·26-s + 0.464·27-s + 0.834·28-s − 0.0318·29-s + 1.12·31-s − 0.176·32-s + 0.101·33-s − 0.171·34-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

\(L(1)\) \(\approx\) \(1.239160250\)
\(L(\frac12)\) \(\approx\) \(1.239160250\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.414T + 3T^{2} \)
7 \( 1 - 4.41T + 7T^{2} \)
11 \( 1 + 1.41T + 11T^{2} \)
13 \( 1 - 5.82T + 13T^{2} \)
17 \( 1 - T + 17T^{2} \)
23 \( 1 - 0.757T + 23T^{2} \)
29 \( 1 + 0.171T + 29T^{2} \)
31 \( 1 - 6.24T + 31T^{2} \)
37 \( 1 + 8.48T + 37T^{2} \)
41 \( 1 + 4.24T + 41T^{2} \)
43 \( 1 - 1.75T + 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 - 5.48T + 53T^{2} \)
59 \( 1 - 6.89T + 59T^{2} \)
61 \( 1 - 14.2T + 61T^{2} \)
67 \( 1 - 4.75T + 67T^{2} \)
71 \( 1 + 13.4T + 71T^{2} \)
73 \( 1 - 11.4T + 73T^{2} \)
79 \( 1 + 6.48T + 79T^{2} \)
83 \( 1 - 14.4T + 83T^{2} \)
89 \( 1 - 7.07T + 89T^{2} \)
97 \( 1 + 0.343T + 97T^{2} \)
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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

−10.28077010141182019098845748019, −8.831862100813789181510536703591, −8.458968035015643325998359679496, −7.85092908730108193919178449370, −6.69205257398238111475743537251, −5.70804418276667345505510457016, −4.97743752742474768646814718854, −3.63150975539577213875955977149, −2.24874253997621326318976801540, −1.04897719567186889567602968690, 1.04897719567186889567602968690, 2.24874253997621326318976801540, 3.63150975539577213875955977149, 4.97743752742474768646814718854, 5.70804418276667345505510457016, 6.69205257398238111475743537251, 7.85092908730108193919178449370, 8.458968035015643325998359679496, 8.831862100813789181510536703591, 10.28077010141182019098845748019

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