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Properties

Label 2-950-95.9-c1-0-4
Degree $2$
Conductor $950$
Sign $0.995 - 0.0997i$
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 + (−1.68 − 2.00i)3-s + (0.939 − 0.342i)4-s + (2.00 + 1.68i)6-s + (−0.840 − 0.485i)7-s + (−0.866 + 0.5i)8-s + (−0.668 + 3.79i)9-s + (0.280 + 0.486i)11-s + (−2.26 − 1.30i)12-s + (−0.293 + 0.350i)13-s + (0.911 + 0.331i)14-s + (0.766 − 0.642i)16-s + (−0.387 + 0.0682i)17-s − 3.84i·18-s + (1.58 + 4.05i)19-s + ⋯
L(s)  = 1  + (−0.696 + 0.122i)2-s + (−0.971 − 1.15i)3-s + (0.469 − 0.171i)4-s + (0.818 + 0.686i)6-s + (−0.317 − 0.183i)7-s + (−0.306 + 0.176i)8-s + (−0.222 + 1.26i)9-s + (0.0846 + 0.146i)11-s + (−0.654 − 0.377i)12-s + (−0.0815 + 0.0971i)13-s + (0.243 + 0.0887i)14-s + (0.191 − 0.160i)16-s + (−0.0938 + 0.0165i)17-s − 0.907i·18-s + (0.364 + 0.931i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 - 0.0997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.995 - 0.0997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $0.995 - 0.0997i$
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: \((2,\ 950,\ (\ :1/2),\ 0.995 - 0.0997i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.613054 + 0.0306594i\)
\(L(\frac12)\) \(\approx\) \(0.613054 + 0.0306594i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.58 - 4.05i)T \)
good3 \( 1 + (1.68 + 2.00i)T + (-0.520 + 2.95i)T^{2} \)
7 \( 1 + (0.840 + 0.485i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.280 - 0.486i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.293 - 0.350i)T + (-2.25 - 12.8i)T^{2} \)
17 \( 1 + (0.387 - 0.0682i)T + (15.9 - 5.81i)T^{2} \)
23 \( 1 + (-1.79 - 4.92i)T + (-17.6 + 14.7i)T^{2} \)
29 \( 1 + (-0.411 + 2.33i)T + (-27.2 - 9.91i)T^{2} \)
31 \( 1 + (5.44 - 9.42i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 5.14iT - 37T^{2} \)
41 \( 1 + (-6.14 + 5.15i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 + (3.49 - 9.61i)T + (-32.9 - 27.6i)T^{2} \)
47 \( 1 + (-12.0 - 2.13i)T + (44.1 + 16.0i)T^{2} \)
53 \( 1 + (1.55 + 4.26i)T + (-40.6 + 34.0i)T^{2} \)
59 \( 1 + (0.759 + 4.30i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (-4.02 + 1.46i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (-12.0 - 2.11i)T + (62.9 + 22.9i)T^{2} \)
71 \( 1 + (-6.85 - 2.49i)T + (54.3 + 45.6i)T^{2} \)
73 \( 1 + (-8.96 - 10.6i)T + (-12.6 + 71.8i)T^{2} \)
79 \( 1 + (-0.730 + 0.612i)T + (13.7 - 77.7i)T^{2} \)
83 \( 1 + (9.76 + 5.63i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (2.69 + 2.26i)T + (15.4 + 87.6i)T^{2} \)
97 \( 1 + (7.10 - 1.25i)T + (91.1 - 33.1i)T^{2} \)
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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

−10.05173019066443176839945884609, −9.259621338951337429263990386169, −8.179190553293884593105841018791, −7.33022672358142650455740436898, −6.86499006000977933895824723876, −5.93306797673555383840496003388, −5.24476081750227522570546847993, −3.60324778061446157644621277528, −2.02935371979510590120517239902, −0.966130828901204099512006935700, 0.53248904312195913531307304645, 2.54778478452182743708671158544, 3.79598455514852455721331457470, 4.78559077208684935682617133927, 5.69564912826744026423382995115, 6.51821994075325873328711764483, 7.50574307050991779538631797940, 8.704602234769257966536989728855, 9.390204016182882046789635620841, 10.00071090512257330232829789418

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