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Properties

Label 2-950-19.7-c1-0-15
Degree $2$
Conductor $950$
Sign $-0.221 + 0.975i$
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  + (−0.5 − 0.866i)2-s + (−0.664 − 1.15i)3-s + (−0.499 + 0.866i)4-s + (−0.664 + 1.15i)6-s − 2.32·7-s + 0.999·8-s + (0.616 − 1.06i)9-s + 6.39·11-s + 1.32·12-s + (−0.429 + 0.743i)13-s + (1.16 + 2.01i)14-s + (−0.5 − 0.866i)16-s + (2.34 + 4.06i)17-s − 1.23·18-s + (3.75 + 2.21i)19-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.383 − 0.664i)3-s + (−0.249 + 0.433i)4-s + (−0.271 + 0.469i)6-s − 0.880·7-s + 0.353·8-s + (0.205 − 0.355i)9-s + 1.92·11-s + 0.383·12-s + (−0.119 + 0.206i)13-s + (0.311 + 0.539i)14-s + (−0.125 − 0.216i)16-s + (0.568 + 0.985i)17-s − 0.290·18-s + (0.860 + 0.509i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.221 + 0.975i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.221 + 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $-0.221 + 0.975i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (501, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ -0.221 + 0.975i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.707596 - 0.886363i\)
\(L(\frac12)\) \(\approx\) \(0.707596 - 0.886363i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 + 0.866i)T \)
5 \( 1 \)
19 \( 1 + (-3.75 - 2.21i)T \)
good3 \( 1 + (0.664 + 1.15i)T + (-1.5 + 2.59i)T^{2} \)
7 \( 1 + 2.32T + 7T^{2} \)
11 \( 1 - 6.39T + 11T^{2} \)
13 \( 1 + (0.429 - 0.743i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2.34 - 4.06i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (-1.73 + 3.00i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.21 + 3.82i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 8.25T + 31T^{2} \)
37 \( 1 - 9.76T + 37T^{2} \)
41 \( 1 + (1.84 + 3.19i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.56 + 6.17i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3.59 + 6.22i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-3.10 + 5.37i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.09 - 5.35i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.01 + 6.94i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.45 + 4.24i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (1.10 + 1.90i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-2.32 - 4.03i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (5.79 + 10.0i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 6.07T + 83T^{2} \)
89 \( 1 + (5.64 - 9.78i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-5.67 - 9.82i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.631519146175217847797250875454, −9.289918592516277336506470607371, −8.224493629783686758745004218951, −7.08730719355722403980478700752, −6.54901645766415467793415252167, −5.69981293151174283884697345229, −4.01326985980045368255700045302, −3.50379208678512685292058752622, −1.85860802354086149075259747756, −0.803628700261515346244694529027, 1.16438413315450881630524331328, 3.14595106809853061389356045402, 4.18823243861168502946957270320, 5.15305480631211983284809331285, 6.00862640297876360983006241679, 6.94692540853130454079853094040, 7.53362540538182589174229862626, 8.880329967427364126458837084941, 9.701175661941259452051508838605, 9.710125076867203109153913987615

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