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Properties

Label 2-950-95.9-c1-0-8
Degree $2$
Conductor $950$
Sign $-0.497 - 0.867i$
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  + (−0.984 + 0.173i)2-s + (2.20 + 2.62i)3-s + (0.939 − 0.342i)4-s + (−2.62 − 2.20i)6-s + (1.61 + 0.933i)7-s + (−0.866 + 0.5i)8-s + (−1.52 + 8.62i)9-s + (1.80 + 3.12i)11-s + (2.96 + 1.71i)12-s + (1.82 − 2.17i)13-s + (−1.75 − 0.638i)14-s + (0.766 − 0.642i)16-s + (5.89 − 1.03i)17-s − 8.75i·18-s + (−4.32 − 0.577i)19-s + ⋯
L(s)  = 1  + (−0.696 + 0.122i)2-s + (1.27 + 1.51i)3-s + (0.469 − 0.171i)4-s + (−1.07 − 0.899i)6-s + (0.611 + 0.352i)7-s + (−0.306 + 0.176i)8-s + (−0.506 + 2.87i)9-s + (0.543 + 0.941i)11-s + (0.857 + 0.494i)12-s + (0.505 − 0.602i)13-s + (−0.468 − 0.170i)14-s + (0.191 − 0.160i)16-s + (1.43 − 0.252i)17-s − 2.06i·18-s + (−0.991 − 0.132i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.497 - 0.867i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.497 - 0.867i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $-0.497 - 0.867i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ -0.497 - 0.867i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.977928 + 1.68895i\)
\(L(\frac12)\) \(\approx\) \(0.977928 + 1.68895i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.984 - 0.173i)T \)
5 \( 1 \)
19 \( 1 + (4.32 + 0.577i)T \)
good3 \( 1 + (-2.20 - 2.62i)T + (-0.520 + 2.95i)T^{2} \)
7 \( 1 + (-1.61 - 0.933i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.80 - 3.12i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.82 + 2.17i)T + (-2.25 - 12.8i)T^{2} \)
17 \( 1 + (-5.89 + 1.03i)T + (15.9 - 5.81i)T^{2} \)
23 \( 1 + (1.74 + 4.78i)T + (-17.6 + 14.7i)T^{2} \)
29 \( 1 + (0.204 - 1.16i)T + (-27.2 - 9.91i)T^{2} \)
31 \( 1 + (-2.59 + 4.50i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 3.35iT - 37T^{2} \)
41 \( 1 + (2.85 - 2.39i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 + (-0.229 + 0.631i)T + (-32.9 - 27.6i)T^{2} \)
47 \( 1 + (-6.22 - 1.09i)T + (44.1 + 16.0i)T^{2} \)
53 \( 1 + (0.876 + 2.40i)T + (-40.6 + 34.0i)T^{2} \)
59 \( 1 + (-0.827 - 4.69i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (7.73 - 2.81i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (7.84 + 1.38i)T + (62.9 + 22.9i)T^{2} \)
71 \( 1 + (-1.81 - 0.659i)T + (54.3 + 45.6i)T^{2} \)
73 \( 1 + (3.22 + 3.83i)T + (-12.6 + 71.8i)T^{2} \)
79 \( 1 + (-0.460 + 0.386i)T + (13.7 - 77.7i)T^{2} \)
83 \( 1 + (-1.64 - 0.951i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (0.755 + 0.633i)T + (15.4 + 87.6i)T^{2} \)
97 \( 1 + (0.184 - 0.0325i)T + (91.1 - 33.1i)T^{2} \)
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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

−10.18663512345175321329281658423, −9.437434342102738361328323810484, −8.710829335315645906498653126531, −8.142106767372828205494762841834, −7.42557635289732359185795933458, −5.89530093307981707143050318804, −4.84671506657073883142885810279, −4.03652872400851563987061723452, −2.92095699360144824743122499028, −1.91999391609921005175146558068, 1.10392268643678634725345265511, 1.76067672096983039571425785841, 3.09125005519615844910665230915, 3.86972559932921095849378531966, 5.93805364694547769299678059147, 6.61356206134885161504207882275, 7.55996968147647442417835418918, 8.100345262562938736707357435423, 8.717629660022014242729646525585, 9.386000464106061550970381503272

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