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Properties

Label 2-950-95.74-c1-0-27
Degree $2$
Conductor $950$
Sign $0.536 + 0.843i$
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 + (1.49 − 1.77i)3-s + (0.939 + 0.342i)4-s + (1.77 − 1.49i)6-s + (4.25 − 2.45i)7-s + (0.866 + 0.5i)8-s + (−0.413 − 2.34i)9-s + (−1.42 + 2.46i)11-s + (2.00 − 1.15i)12-s + (−4.21 − 5.02i)13-s + (4.62 − 1.68i)14-s + (0.766 + 0.642i)16-s + (2.15 + 0.380i)17-s − 2.38i·18-s + (−4.17 + 1.26i)19-s + ⋯
L(s)  = 1  + (0.696 + 0.122i)2-s + (0.860 − 1.02i)3-s + (0.469 + 0.171i)4-s + (0.725 − 0.608i)6-s + (1.60 − 0.929i)7-s + (0.306 + 0.176i)8-s + (−0.137 − 0.781i)9-s + (−0.429 + 0.743i)11-s + (0.579 − 0.334i)12-s + (−1.16 − 1.39i)13-s + (1.23 − 0.449i)14-s + (0.191 + 0.160i)16-s + (0.523 + 0.0922i)17-s − 0.561i·18-s + (−0.957 + 0.289i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.536 + 0.843i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.536 + 0.843i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $0.536 + 0.843i$
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: \((2,\ 950,\ (\ :1/2),\ 0.536 + 0.843i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.07758 - 1.68960i\)
\(L(\frac12)\) \(\approx\) \(3.07758 - 1.68960i\)
\(L(\frac{3}{2})\) 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 \( 1 + (4.17 - 1.26i)T \)
good3 \( 1 + (-1.49 + 1.77i)T + (-0.520 - 2.95i)T^{2} \)
7 \( 1 + (-4.25 + 2.45i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.42 - 2.46i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (4.21 + 5.02i)T + (-2.25 + 12.8i)T^{2} \)
17 \( 1 + (-2.15 - 0.380i)T + (15.9 + 5.81i)T^{2} \)
23 \( 1 + (0.908 - 2.49i)T + (-17.6 - 14.7i)T^{2} \)
29 \( 1 + (-1.32 - 7.48i)T + (-27.2 + 9.91i)T^{2} \)
31 \( 1 + (-2.70 - 4.68i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 2.06iT - 37T^{2} \)
41 \( 1 + (7.16 + 6.01i)T + (7.11 + 40.3i)T^{2} \)
43 \( 1 + (0.593 + 1.62i)T + (-32.9 + 27.6i)T^{2} \)
47 \( 1 + (2.95 - 0.521i)T + (44.1 - 16.0i)T^{2} \)
53 \( 1 + (1.44 - 3.96i)T + (-40.6 - 34.0i)T^{2} \)
59 \( 1 + (-0.167 + 0.949i)T + (-55.4 - 20.1i)T^{2} \)
61 \( 1 + (-7.04 - 2.56i)T + (46.7 + 39.2i)T^{2} \)
67 \( 1 + (-1.89 + 0.333i)T + (62.9 - 22.9i)T^{2} \)
71 \( 1 + (-5.79 + 2.10i)T + (54.3 - 45.6i)T^{2} \)
73 \( 1 + (-2.55 + 3.04i)T + (-12.6 - 71.8i)T^{2} \)
79 \( 1 + (11.7 + 9.84i)T + (13.7 + 77.7i)T^{2} \)
83 \( 1 + (1.13 - 0.656i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (-8.25 + 6.92i)T + (15.4 - 87.6i)T^{2} \)
97 \( 1 + (2.02 + 0.356i)T + (91.1 + 33.1i)T^{2} \)
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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

−10.28470731562041038529808367196, −8.594262052513855375517201404346, −7.921352388839380039377572914788, −7.49254930233704258144001472344, −6.83684262176739264212908484556, −5.25969401105682922822993343200, −4.77511562833359383086211525766, −3.44046926252907167312164149874, −2.30655908522127804253770808336, −1.40430762881518538807800568729, 2.08530689672418123428960420807, 2.72057560461246307478786095690, 4.16155353011144467387657740244, 4.64803059602477668979733784514, 5.46930336616146488263759482659, 6.63688578043787889324685445420, 8.066440518769115525279193860301, 8.378973461806962339994196407055, 9.411476448087396559345983844275, 10.07983240479857888186155806582

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