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Properties

Label 2-950-95.74-c1-0-0
Degree $2$
Conductor $950$
Sign $-0.352 + 0.935i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.352 + 0.935i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.352 + 0.935i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $-0.352 + 0.935i$
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: \((2,\ 950,\ (\ :1/2),\ -0.352 + 0.935i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.129116 - 0.186689i\)
\(L(\frac12)\) \(\approx\) \(0.129116 - 0.186689i\)
\(L(\frac{3}{2})\) 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

−10.41945677444133884440059213876, −9.926510164493248040770777938498, −8.949565640160618820999137310760, −8.712578243457485485331393649179, −6.88131332878730871904382001959, −6.47393329862048816469554215141, −5.54000778353327513988758219977, −4.41954859330731688556907582191, −3.42131439637230178526304671642, −2.10825273844134223681013575189, 0.18673542541989125746331381619, 0.957517143884764233610221433629, 2.76718482053492086875035241916, 3.85428314303348109728341656805, 5.65857417774296213867982850878, 6.28319110682168649279842627532, 6.72919609869602745218045360098, 7.74141783380930215331662817681, 8.420384954057088523320909912593, 9.584694420395657753609962891285

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