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Properties

Label 2-950-95.74-c1-0-17
Degree $2$
Conductor $950$
Sign $0.999 - 0.0173i$
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 + (0.565 − 0.673i)3-s + (0.939 + 0.342i)4-s + (0.673 − 0.565i)6-s + (0.460 − 0.266i)7-s + (0.866 + 0.5i)8-s + (0.386 + 2.19i)9-s + (0.5 − 0.866i)11-s + (0.761 − 0.439i)12-s + (−0.419 − 0.5i)13-s + (0.5 − 0.181i)14-s + (0.766 + 0.642i)16-s + (3.82 + 0.673i)17-s + 2.22i·18-s + (4.21 + 1.10i)19-s + ⋯
L(s)  = 1  + (0.696 + 0.122i)2-s + (0.326 − 0.388i)3-s + (0.469 + 0.171i)4-s + (0.275 − 0.230i)6-s + (0.174 − 0.100i)7-s + (0.306 + 0.176i)8-s + (0.128 + 0.730i)9-s + (0.150 − 0.261i)11-s + (0.219 − 0.126i)12-s + (−0.116 − 0.138i)13-s + (0.133 − 0.0486i)14-s + (0.191 + 0.160i)16-s + (0.926 + 0.163i)17-s + 0.524i·18-s + (0.967 + 0.253i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0173i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0173i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $0.999 - 0.0173i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (549, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ 0.999 - 0.0173i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.90287 + 0.0251716i\)
\(L(\frac12)\) \(\approx\) \(2.90287 + 0.0251716i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-4.21 - 1.10i)T \)
good3 \( 1 + (-0.565 + 0.673i)T + (-0.520 - 2.95i)T^{2} \)
7 \( 1 + (-0.460 + 0.266i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.5 + 0.866i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.419 + 0.5i)T + (-2.25 + 12.8i)T^{2} \)
17 \( 1 + (-3.82 - 0.673i)T + (15.9 + 5.81i)T^{2} \)
23 \( 1 + (-1.55 + 4.25i)T + (-17.6 - 14.7i)T^{2} \)
29 \( 1 + (-0.773 - 4.38i)T + (-27.2 + 9.91i)T^{2} \)
31 \( 1 + (1.37 + 2.38i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 0.958iT - 37T^{2} \)
41 \( 1 + (4.78 + 4.01i)T + (7.11 + 40.3i)T^{2} \)
43 \( 1 + (1.71 + 4.70i)T + (-32.9 + 27.6i)T^{2} \)
47 \( 1 + (0.761 - 0.134i)T + (44.1 - 16.0i)T^{2} \)
53 \( 1 + (-1.31 + 3.62i)T + (-40.6 - 34.0i)T^{2} \)
59 \( 1 + (1.08 - 6.17i)T + (-55.4 - 20.1i)T^{2} \)
61 \( 1 + (4.62 + 1.68i)T + (46.7 + 39.2i)T^{2} \)
67 \( 1 + (1.32 - 0.233i)T + (62.9 - 22.9i)T^{2} \)
71 \( 1 + (5.33 - 1.94i)T + (54.3 - 45.6i)T^{2} \)
73 \( 1 + (-4.80 + 5.72i)T + (-12.6 - 71.8i)T^{2} \)
79 \( 1 + (-0.543 - 0.455i)T + (13.7 + 77.7i)T^{2} \)
83 \( 1 + (-9.15 + 5.28i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (3.75 - 3.15i)T + (15.4 - 87.6i)T^{2} \)
97 \( 1 + (-0.0994 - 0.0175i)T + (91.1 + 33.1i)T^{2} \)
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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

−10.26134918184494908209608097241, −9.086409813309042735702117894315, −8.106771512053797807352169000258, −7.51837511848826652423937730993, −6.67933422278929795565178867385, −5.55856175820453546320098298768, −4.88192367376312120399072150983, −3.66324967037578509759801637062, −2.70475863808577121762547077485, −1.43771827636360780141038246171, 1.36304858170247337567003253904, 2.92727558758543307542264881885, 3.64364129891689547875860825356, 4.69526874114820710312643999660, 5.52492215395235256366770249090, 6.55801460138902862981261402488, 7.40597991489111891896576460680, 8.356350840574728603581497098426, 9.584630799566032945165160017749, 9.765328280715746757163685854351

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