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Properties

Label 2-950-95.64-c1-0-25
Degree $2$
Conductor $950$
Sign $-0.0242 + 0.999i$
Analytic cond. $7.58578$
Root an. cond. $2.75423$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.866 − 0.5i)2-s + (2.29 − 1.32i)3-s + (0.499 − 0.866i)4-s + (1.32 − 2.29i)6-s − 1.64i·7-s − 0.999i·8-s + (2 − 3.46i)9-s + 0.645·11-s − 2.64i·12-s + (1.73 + i)13-s + (−0.822 − 1.42i)14-s + (−0.5 − 0.866i)16-s − 3.99i·18-s + (−4.32 + 0.559i)19-s + (−2.17 − 3.77i)21-s + (0.559 − 0.322i)22-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (1.32 − 0.763i)3-s + (0.249 − 0.433i)4-s + (0.540 − 0.935i)6-s − 0.622i·7-s − 0.353i·8-s + (0.666 − 1.15i)9-s + 0.194·11-s − 0.763i·12-s + (0.480 + 0.277i)13-s + (−0.219 − 0.380i)14-s + (−0.125 − 0.216i)16-s − 0.942i·18-s + (−0.991 + 0.128i)19-s + (−0.475 − 0.822i)21-s + (0.119 − 0.0688i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0242 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0242 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $-0.0242 + 0.999i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ -0.0242 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.42429 - 2.48381i\)
\(L(\frac12)\) \(\approx\) \(2.42429 - 2.48381i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 + 0.5i)T \)
5 \( 1 \)
19 \( 1 + (4.32 - 0.559i)T \)
good3 \( 1 + (-2.29 + 1.32i)T + (1.5 - 2.59i)T^{2} \)
7 \( 1 + 1.64iT - 7T^{2} \)
11 \( 1 - 0.645T + 11T^{2} \)
13 \( 1 + (-1.73 - i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (8.5 - 14.7i)T^{2} \)
23 \( 1 + (-3.15 - 1.82i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.82 - 3.15i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 0.354T + 31T^{2} \)
37 \( 1 - 5.64iT - 37T^{2} \)
41 \( 1 + (5.14 + 8.91i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.613 - 0.354i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (8.35 + 4.82i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-7.43 - 4.29i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3.96 - 6.87i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-7.46 + 12.9i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.02 - 2.32i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-6.64 - 11.5i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (10.6 - 6.14i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (2 + 3.46i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 7.93iT - 83T^{2} \)
89 \( 1 + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (12.3 - 7.14i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.814856248003402532771255201275, −8.812801281996926499777596118730, −8.273845184456216602768229036068, −7.10910837707285450677556807031, −6.73621078656929129144314494396, −5.39390415175583565445568121561, −4.08575858379804046659958144721, −3.42248623021053413759725013182, −2.30946247568206623010226832340, −1.31416053113983390625556210273, 2.15479086242211216012008716380, 3.07404633784183368897651110187, 3.93897393542585119773622167173, 4.77992932715118259003758225576, 5.86032065697631684154382728683, 6.83132356586465129743674869435, 7.993586079639861846637149731731, 8.562155259424626829518065629514, 9.205374912043458360735131336408, 10.09696780350193346209736636315

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