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Properties

Label 2-950-95.53-c1-0-11
Degree $2$
Conductor $950$
Sign $0.393 - 0.919i$
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  + (−0.422 − 0.906i)2-s + (2.64 + 1.85i)3-s + (−0.642 + 0.766i)4-s + (0.560 − 3.17i)6-s + (0.958 + 3.57i)7-s + (0.965 + 0.258i)8-s + (2.53 + 6.96i)9-s + (−1.03 − 1.78i)11-s + (−3.11 + 0.835i)12-s + (1.25 + 1.78i)13-s + (2.83 − 2.37i)14-s + (−0.173 − 0.984i)16-s + (−1.16 + 0.543i)17-s + (5.24 − 5.24i)18-s + (−1.10 − 4.21i)19-s + ⋯
L(s)  = 1  + (−0.298 − 0.640i)2-s + (1.52 + 1.06i)3-s + (−0.321 + 0.383i)4-s + (0.228 − 1.29i)6-s + (0.362 + 1.35i)7-s + (0.341 + 0.0915i)8-s + (0.844 + 2.32i)9-s + (−0.311 − 0.538i)11-s + (−0.899 + 0.241i)12-s + (0.347 + 0.496i)13-s + (0.757 − 0.635i)14-s + (−0.0434 − 0.246i)16-s + (−0.282 + 0.131i)17-s + (1.23 − 1.23i)18-s + (−0.253 − 0.967i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.393 - 0.919i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.393 - 0.919i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $0.393 - 0.919i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (243, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ 0.393 - 0.919i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.85084 + 1.22082i\)
\(L(\frac12)\) \(\approx\) \(1.85084 + 1.22082i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.422 + 0.906i)T \)
5 \( 1 \)
19 \( 1 + (1.10 + 4.21i)T \)
good3 \( 1 + (-2.64 - 1.85i)T + (1.02 + 2.81i)T^{2} \)
7 \( 1 + (-0.958 - 3.57i)T + (-6.06 + 3.5i)T^{2} \)
11 \( 1 + (1.03 + 1.78i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.25 - 1.78i)T + (-4.44 + 12.2i)T^{2} \)
17 \( 1 + (1.16 - 0.543i)T + (10.9 - 13.0i)T^{2} \)
23 \( 1 + (0.322 + 3.68i)T + (-22.6 + 3.99i)T^{2} \)
29 \( 1 + (7.40 - 2.69i)T + (22.2 - 18.6i)T^{2} \)
31 \( 1 + (-7.91 - 4.56i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (4.90 + 4.90i)T + 37iT^{2} \)
41 \( 1 + (7.67 - 1.35i)T + (38.5 - 14.0i)T^{2} \)
43 \( 1 + (-9.90 - 0.866i)T + (42.3 + 7.46i)T^{2} \)
47 \( 1 + (-4.11 + 8.82i)T + (-30.2 - 36.0i)T^{2} \)
53 \( 1 + (-9.05 + 0.791i)T + (52.1 - 9.20i)T^{2} \)
59 \( 1 + (1.99 + 0.726i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (-8.62 - 7.23i)T + (10.5 + 60.0i)T^{2} \)
67 \( 1 + (-2.31 - 1.07i)T + (43.0 + 51.3i)T^{2} \)
71 \( 1 + (2.53 + 3.01i)T + (-12.3 + 69.9i)T^{2} \)
73 \( 1 + (-3.03 + 4.33i)T + (-24.9 - 68.5i)T^{2} \)
79 \( 1 + (-0.385 - 2.18i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (13.7 - 3.67i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 + (-0.335 + 1.90i)T + (-83.6 - 30.4i)T^{2} \)
97 \( 1 + (-2.17 - 4.66i)T + (-62.3 + 74.3i)T^{2} \)
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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

−10.10251101365778475809049161778, −9.105029073816754791029955700280, −8.708204845327436761569047279786, −8.380434272568567574644361160449, −7.12122538116345655459381625363, −5.51939000241232930543660698863, −4.62034467121148202250103927914, −3.66666628511628926537260201823, −2.68438383072480622320217195114, −2.07037283739361723828984194654, 1.01167993943986881201465255467, 2.10315292995484860497803469927, 3.51963969147289825421372933221, 4.32669283334444982021339770605, 5.89240020323815139393292528290, 6.93547368425529552451150334324, 7.54451824275445431159258301824, 7.968666825658126126858577264044, 8.730590292293151126408762946098, 9.741181421721520588405688937559

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