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Properties

Label 2-950-19.4-c1-0-15
Degree $2$
Conductor $950$
Sign $0.960 + 0.278i$
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  + (−0.939 − 0.342i)2-s + (−0.00627 − 0.0355i)3-s + (0.766 + 0.642i)4-s + (−0.00627 + 0.0355i)6-s + (0.918 − 1.59i)7-s + (−0.500 − 0.866i)8-s + (2.81 − 1.02i)9-s + (1.23 + 2.13i)11-s + (0.0180 − 0.0313i)12-s + (−0.415 + 2.35i)13-s + (−1.40 + 1.18i)14-s + (0.173 + 0.984i)16-s + (6.33 + 2.30i)17-s − 2.99·18-s + (−4.34 − 0.298i)19-s + ⋯
L(s)  = 1  + (−0.664 − 0.241i)2-s + (−0.00362 − 0.0205i)3-s + (0.383 + 0.321i)4-s + (−0.00256 + 0.0145i)6-s + (0.347 − 0.601i)7-s + (−0.176 − 0.306i)8-s + (0.939 − 0.341i)9-s + (0.371 + 0.643i)11-s + (0.00521 − 0.00903i)12-s + (−0.115 + 0.653i)13-s + (−0.376 + 0.315i)14-s + (0.0434 + 0.246i)16-s + (1.53 + 0.559i)17-s − 0.706·18-s + (−0.997 − 0.0684i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.960 + 0.278i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.960 + 0.278i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $0.960 + 0.278i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ 0.960 + 0.278i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.32579 - 0.188190i\)
\(L(\frac12)\) \(\approx\) \(1.32579 - 0.188190i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.939 + 0.342i)T \)
5 \( 1 \)
19 \( 1 + (4.34 + 0.298i)T \)
good3 \( 1 + (0.00627 + 0.0355i)T + (-2.81 + 1.02i)T^{2} \)
7 \( 1 + (-0.918 + 1.59i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.23 - 2.13i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.415 - 2.35i)T + (-12.2 - 4.44i)T^{2} \)
17 \( 1 + (-6.33 - 2.30i)T + (13.0 + 10.9i)T^{2} \)
23 \( 1 + (1.24 + 1.04i)T + (3.99 + 22.6i)T^{2} \)
29 \( 1 + (3.10 - 1.12i)T + (22.2 - 18.6i)T^{2} \)
31 \( 1 + (-1.75 + 3.03i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 6.00T + 37T^{2} \)
41 \( 1 + (1.38 + 7.85i)T + (-38.5 + 14.0i)T^{2} \)
43 \( 1 + (4.37 - 3.67i)T + (7.46 - 42.3i)T^{2} \)
47 \( 1 + (-11.7 + 4.27i)T + (36.0 - 30.2i)T^{2} \)
53 \( 1 + (-1.47 - 1.23i)T + (9.20 + 52.1i)T^{2} \)
59 \( 1 + (-4.46 - 1.62i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (-10.4 - 8.78i)T + (10.5 + 60.0i)T^{2} \)
67 \( 1 + (3.49 - 1.27i)T + (51.3 - 43.0i)T^{2} \)
71 \( 1 + (-4.46 + 3.74i)T + (12.3 - 69.9i)T^{2} \)
73 \( 1 + (-0.0886 - 0.502i)T + (-68.5 + 24.9i)T^{2} \)
79 \( 1 + (-2.10 - 11.9i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (-3.11 + 5.39i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-2.95 + 16.7i)T + (-83.6 - 30.4i)T^{2} \)
97 \( 1 + (12.8 + 4.67i)T + (74.3 + 62.3i)T^{2} \)
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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

−10.05661399603169247416067535487, −9.317299127998050875916540841124, −8.358881081928053521119682368396, −7.43756272840950823231991232741, −6.93250498110412302884497531540, −5.84241331463735731863957052776, −4.36436676870944511160495022855, −3.81575025913778670692742603460, −2.14936662772608288718291075024, −1.10021157919549265225798612712, 1.07021036770647557073589853335, 2.38542548235252486939949153277, 3.70242338939580964266364977271, 5.03447098962028286428912880822, 5.80446163191900821953360355822, 6.78762308595600264746203135121, 7.79713763450527520753946456821, 8.256851832071906848269987163797, 9.287132260811256704939735852645, 9.996556452245496213264810246128

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