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Properties

Label 2-950-95.44-c1-0-18
Degree $2$
Conductor $950$
Sign $0.991 - 0.128i$
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  + (0.642 + 0.766i)2-s + (0.931 − 2.55i)3-s + (−0.173 + 0.984i)4-s + (2.55 − 0.931i)6-s + (2.31 + 1.33i)7-s + (−0.866 + 0.500i)8-s + (−3.37 − 2.83i)9-s + (2.46 + 4.26i)11-s + (2.35 + 1.36i)12-s + (1.54 + 4.24i)13-s + (0.463 + 2.62i)14-s + (−0.939 − 0.342i)16-s + (3.15 + 3.76i)17-s − 4.41i·18-s + (0.628 − 4.31i)19-s + ⋯
L(s)  = 1  + (0.454 + 0.541i)2-s + (0.537 − 1.47i)3-s + (−0.0868 + 0.492i)4-s + (1.04 − 0.380i)6-s + (0.873 + 0.504i)7-s + (−0.306 + 0.176i)8-s + (−1.12 − 0.945i)9-s + (0.742 + 1.28i)11-s + (0.680 + 0.392i)12-s + (0.428 + 1.17i)13-s + (0.123 + 0.702i)14-s + (−0.234 − 0.0855i)16-s + (0.765 + 0.911i)17-s − 1.03i·18-s + (0.144 − 0.989i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.128i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 - 0.128i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $0.991 - 0.128i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (899, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ 0.991 - 0.128i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.69899 + 0.174732i\)
\(L(\frac12)\) \(\approx\) \(2.69899 + 0.174732i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.642 - 0.766i)T \)
5 \( 1 \)
19 \( 1 + (-0.628 + 4.31i)T \)
good3 \( 1 + (-0.931 + 2.55i)T + (-2.29 - 1.92i)T^{2} \)
7 \( 1 + (-2.31 - 1.33i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.46 - 4.26i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.54 - 4.24i)T + (-9.95 + 8.35i)T^{2} \)
17 \( 1 + (-3.15 - 3.76i)T + (-2.95 + 16.7i)T^{2} \)
23 \( 1 + (3.07 + 0.542i)T + (21.6 + 7.86i)T^{2} \)
29 \( 1 + (0.227 + 0.190i)T + (5.03 + 28.5i)T^{2} \)
31 \( 1 + (-3.13 + 5.43i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 8.79iT - 37T^{2} \)
41 \( 1 + (7.15 + 2.60i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (2.99 - 0.528i)T + (40.4 - 14.7i)T^{2} \)
47 \( 1 + (1.45 - 1.73i)T + (-8.16 - 46.2i)T^{2} \)
53 \( 1 + (13.0 + 2.29i)T + (49.8 + 18.1i)T^{2} \)
59 \( 1 + (-9.63 + 8.08i)T + (10.2 - 58.1i)T^{2} \)
61 \( 1 + (0.806 - 4.57i)T + (-57.3 - 20.8i)T^{2} \)
67 \( 1 + (6.27 - 7.47i)T + (-11.6 - 65.9i)T^{2} \)
71 \( 1 + (-0.702 - 3.98i)T + (-66.7 + 24.2i)T^{2} \)
73 \( 1 + (1.62 - 4.45i)T + (-55.9 - 46.9i)T^{2} \)
79 \( 1 + (-0.708 - 0.257i)T + (60.5 + 50.7i)T^{2} \)
83 \( 1 + (-9.57 - 5.52i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (-7.44 + 2.71i)T + (68.1 - 57.2i)T^{2} \)
97 \( 1 + (4.57 + 5.45i)T + (-16.8 + 95.5i)T^{2} \)
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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

−9.773129818148955466935003213680, −8.836567775506272770990294927914, −8.205132970620080215282793182951, −7.42126104269309471224210577426, −6.75742256612060127818135214635, −6.04500298214124828216824361442, −4.81986224292137297268034901318, −3.82812944814792602252573082315, −2.25205388436189680816658766082, −1.60426650823427225984253136453, 1.25557984198739581960246887650, 3.27112190009486077722710543922, 3.39944873407657769364520342791, 4.61893661629289584292420671373, 5.27291740554179814339450751150, 6.26208083932333812066439610957, 7.940133352417561821640954453325, 8.423313132554362041856439682396, 9.415431384176166725437826002241, 10.23278898468005361399276315539

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