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Properties

Label 2-950-19.4-c1-0-19
Degree $2$
Conductor $950$
Sign $-0.194 + 0.980i$
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 + (−0.237 − 1.34i)3-s + (0.766 + 0.642i)4-s + (−0.237 + 1.34i)6-s + (−0.816 + 1.41i)7-s + (−0.500 − 0.866i)8-s + (1.05 − 0.384i)9-s + (0.238 + 0.413i)11-s + (0.684 − 1.18i)12-s + (0.712 − 4.03i)13-s + (1.25 − 1.04i)14-s + (0.173 + 0.984i)16-s + (−0.532 − 0.193i)17-s − 1.12·18-s + (0.0784 + 4.35i)19-s + ⋯
L(s)  = 1  + (−0.664 − 0.241i)2-s + (−0.137 − 0.778i)3-s + (0.383 + 0.321i)4-s + (−0.0971 + 0.550i)6-s + (−0.308 + 0.534i)7-s + (−0.176 − 0.306i)8-s + (0.352 − 0.128i)9-s + (0.0719 + 0.124i)11-s + (0.197 − 0.342i)12-s + (0.197 − 1.12i)13-s + (0.334 − 0.280i)14-s + (0.0434 + 0.246i)16-s + (−0.129 − 0.0469i)17-s − 0.264·18-s + (0.0179 + 0.999i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.194 + 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.194 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $-0.194 + 0.980i$
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: \((2,\ 950,\ (\ :1/2),\ -0.194 + 0.980i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.644317 - 0.784428i\)
\(L(\frac12)\) \(\approx\) \(0.644317 - 0.784428i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.0784 - 4.35i)T \)
good3 \( 1 + (0.237 + 1.34i)T + (-2.81 + 1.02i)T^{2} \)
7 \( 1 + (0.816 - 1.41i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.238 - 0.413i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.712 + 4.03i)T + (-12.2 - 4.44i)T^{2} \)
17 \( 1 + (0.532 + 0.193i)T + (13.0 + 10.9i)T^{2} \)
23 \( 1 + (0.893 + 0.750i)T + (3.99 + 22.6i)T^{2} \)
29 \( 1 + (-4.99 + 1.81i)T + (22.2 - 18.6i)T^{2} \)
31 \( 1 + (-4.51 + 7.81i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 6.47T + 37T^{2} \)
41 \( 1 + (0.536 + 3.04i)T + (-38.5 + 14.0i)T^{2} \)
43 \( 1 + (-4.39 + 3.68i)T + (7.46 - 42.3i)T^{2} \)
47 \( 1 + (-2.52 + 0.920i)T + (36.0 - 30.2i)T^{2} \)
53 \( 1 + (7.51 + 6.30i)T + (9.20 + 52.1i)T^{2} \)
59 \( 1 + (6.11 + 2.22i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (9.08 + 7.61i)T + (10.5 + 60.0i)T^{2} \)
67 \( 1 + (4.44 - 1.61i)T + (51.3 - 43.0i)T^{2} \)
71 \( 1 + (6.52 - 5.47i)T + (12.3 - 69.9i)T^{2} \)
73 \( 1 + (-0.219 - 1.24i)T + (-68.5 + 24.9i)T^{2} \)
79 \( 1 + (1.23 + 7.02i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (-3.83 + 6.64i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-3.18 + 18.0i)T + (-83.6 - 30.4i)T^{2} \)
97 \( 1 + (-2.02 - 0.736i)T + (74.3 + 62.3i)T^{2} \)
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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

−9.883972034094490451525058289029, −8.978001989095253938250701625558, −7.958808091053748836113425611297, −7.57710553566437875578765299350, −6.32990679240198864746575511995, −5.91215345889136151574918572542, −4.35688546010755658295112309590, −3.07415286525763575347362764612, −1.99138364032656935357043028191, −0.67801312564160783934457331096, 1.29317695122427817724216647248, 2.90655906893937422742167926260, 4.23059614260235454871390110162, 4.84455882811950669964657513403, 6.22095815794396054964310861724, 6.90054895157156607914237132346, 7.74901225041859313254431986990, 8.909780959515731559623723530328, 9.372081777057974116941443075865, 10.25737057431056273858451013904

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