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Properties

Label 2-950-19.16-c1-0-10
Degree $2$
Conductor $950$
Sign $0.995 + 0.0923i$
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  + (−0.766 − 0.642i)2-s + (−2.56 + 0.934i)3-s + (0.173 + 0.984i)4-s + (2.56 + 0.934i)6-s + (1.06 + 1.85i)7-s + (0.500 − 0.866i)8-s + (3.42 − 2.87i)9-s + (0.926 − 1.60i)11-s + (−1.36 − 2.36i)12-s + (−5.18 − 1.88i)13-s + (0.371 − 2.10i)14-s + (−0.939 + 0.342i)16-s + (2.82 + 2.37i)17-s − 4.46·18-s + (1.21 − 4.18i)19-s + ⋯
L(s)  = 1  + (−0.541 − 0.454i)2-s + (−1.48 + 0.539i)3-s + (0.0868 + 0.492i)4-s + (1.04 + 0.381i)6-s + (0.403 + 0.699i)7-s + (0.176 − 0.306i)8-s + (1.14 − 0.957i)9-s + (0.279 − 0.483i)11-s + (−0.394 − 0.683i)12-s + (−1.43 − 0.522i)13-s + (0.0991 − 0.562i)14-s + (−0.234 + 0.0855i)16-s + (0.686 + 0.575i)17-s − 1.05·18-s + (0.278 − 0.960i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 + 0.0923i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.995 + 0.0923i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $0.995 + 0.0923i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (301, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ 0.995 + 0.0923i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.628407 - 0.0290803i\)
\(L(\frac12)\) \(\approx\) \(0.628407 - 0.0290803i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.766 + 0.642i)T \)
5 \( 1 \)
19 \( 1 + (-1.21 + 4.18i)T \)
good3 \( 1 + (2.56 - 0.934i)T + (2.29 - 1.92i)T^{2} \)
7 \( 1 + (-1.06 - 1.85i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.926 + 1.60i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (5.18 + 1.88i)T + (9.95 + 8.35i)T^{2} \)
17 \( 1 + (-2.82 - 2.37i)T + (2.95 + 16.7i)T^{2} \)
23 \( 1 + (0.511 + 2.89i)T + (-21.6 + 7.86i)T^{2} \)
29 \( 1 + (5.36 - 4.50i)T + (5.03 - 28.5i)T^{2} \)
31 \( 1 + (-2.12 - 3.68i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 3.49T + 37T^{2} \)
41 \( 1 + (-1.47 + 0.536i)T + (31.4 - 26.3i)T^{2} \)
43 \( 1 + (-0.404 + 2.29i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (-8.59 + 7.20i)T + (8.16 - 46.2i)T^{2} \)
53 \( 1 + (-1.11 - 6.32i)T + (-49.8 + 18.1i)T^{2} \)
59 \( 1 + (-10.8 - 9.07i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (0.0461 + 0.261i)T + (-57.3 + 20.8i)T^{2} \)
67 \( 1 + (-11.3 + 9.49i)T + (11.6 - 65.9i)T^{2} \)
71 \( 1 + (-0.722 + 4.09i)T + (-66.7 - 24.2i)T^{2} \)
73 \( 1 + (-14.4 + 5.27i)T + (55.9 - 46.9i)T^{2} \)
79 \( 1 + (-2.70 + 0.985i)T + (60.5 - 50.7i)T^{2} \)
83 \( 1 + (-7.00 - 12.1i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (1.86 + 0.676i)T + (68.1 + 57.2i)T^{2} \)
97 \( 1 + (-8.28 - 6.94i)T + (16.8 + 95.5i)T^{2} \)
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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

−10.28155995319828376608399061090, −9.383404204820458305873203332008, −8.586118096731687947890215601519, −7.47135360787298222168388712072, −6.56691367171163062746500683493, −5.40544108305987504537698102955, −5.06430233036087865168340369865, −3.75190705607989288075196449066, −2.37632222203099904854970826717, −0.69122229959828286234616024135, 0.76625407806191501190598262526, 1.99611163874847538655009328614, 4.13336045983958901136398807148, 5.12610855361865690381200723761, 5.77341394852931520467522953944, 6.83395029475577516340990577087, 7.37006948998311891946376264436, 7.958026806884110298966150937549, 9.603838350112949285075765081080, 9.892183219206808745849764697317

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