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Properties

Label 2-950-25.21-c1-0-26
Degree $2$
Conductor $950$
Sign $0.989 - 0.143i$
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  + (0.309 − 0.951i)2-s + (1.89 + 1.37i)3-s + (−0.809 − 0.587i)4-s + (1.73 + 1.40i)5-s + (1.89 − 1.37i)6-s + 2.10·7-s + (−0.809 + 0.587i)8-s + (0.771 + 2.37i)9-s + (1.87 − 1.21i)10-s + (−0.871 + 2.68i)11-s + (−0.724 − 2.22i)12-s + (−1.13 − 3.49i)13-s + (0.650 − 2.00i)14-s + (1.34 + 5.06i)15-s + (0.309 + 0.951i)16-s + (2.68 − 1.95i)17-s + ⋯
L(s)  = 1  + (0.218 − 0.672i)2-s + (1.09 + 0.795i)3-s + (−0.404 − 0.293i)4-s + (0.776 + 0.630i)5-s + (0.774 − 0.562i)6-s + 0.795·7-s + (−0.286 + 0.207i)8-s + (0.257 + 0.791i)9-s + (0.593 − 0.384i)10-s + (−0.262 + 0.808i)11-s + (−0.209 − 0.643i)12-s + (−0.314 − 0.969i)13-s + (0.173 − 0.535i)14-s + (0.348 + 1.30i)15-s + (0.0772 + 0.237i)16-s + (0.651 − 0.473i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.989 - 0.143i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.989 - 0.143i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $0.989 - 0.143i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (571, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ 0.989 - 0.143i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.93712 + 0.211564i\)
\(L(\frac12)\) \(\approx\) \(2.93712 + 0.211564i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.309 + 0.951i)T \)
5 \( 1 + (-1.73 - 1.40i)T \)
19 \( 1 + (-0.809 + 0.587i)T \)
good3 \( 1 + (-1.89 - 1.37i)T + (0.927 + 2.85i)T^{2} \)
7 \( 1 - 2.10T + 7T^{2} \)
11 \( 1 + (0.871 - 2.68i)T + (-8.89 - 6.46i)T^{2} \)
13 \( 1 + (1.13 + 3.49i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (-2.68 + 1.95i)T + (5.25 - 16.1i)T^{2} \)
23 \( 1 + (-0.636 + 1.95i)T + (-18.6 - 13.5i)T^{2} \)
29 \( 1 + (-2.53 - 1.84i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (4.67 - 3.39i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-2.58 - 7.96i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-1.01 - 3.11i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 7.97T + 43T^{2} \)
47 \( 1 + (6.27 + 4.55i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (3.73 + 2.71i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (1.79 + 5.51i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-0.312 + 0.961i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (-7.18 + 5.21i)T + (20.7 - 63.7i)T^{2} \)
71 \( 1 + (-0.946 - 0.687i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (3.47 - 10.6i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-4.59 - 3.34i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-11.0 + 8.04i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + (-3.96 + 12.1i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (13.5 + 9.81i)T + (29.9 + 92.2i)T^{2} \)
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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

−9.900012855025694606341071569327, −9.655649790849188254396248819201, −8.516754968397360113410204086349, −7.83218086366216701249537937629, −6.68042112818452252018947734930, −5.24095616417799624048207038244, −4.76063544651013524559872414440, −3.36596107138389003812486584521, −2.80672691440404395239122266783, −1.71940006658324542617209138410, 1.38880193637189718190535190683, 2.37844905275967319956049556801, 3.69964725439930186455232379222, 4.91039491368197554064727492159, 5.74163523688787226842615945498, 6.68675660956956222795044084011, 7.78157554038282586469883343451, 8.119031610010606321515083449268, 9.016302392975298278428225862244, 9.526311675188653051456316758060

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