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Properties

Label 2-950-95.4-c1-0-29
Degree $2$
Conductor $950$
Sign $-0.793 - 0.608i$
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  + (0.342 − 0.939i)2-s + (1.31 − 0.231i)3-s + (−0.766 − 0.642i)4-s + (0.231 − 1.31i)6-s + (−3.78 − 2.18i)7-s + (−0.866 + 0.500i)8-s + (−1.15 + 0.419i)9-s + (0.457 + 0.792i)11-s + (−1.15 − 0.665i)12-s + (−3.97 − 0.700i)13-s + (−3.34 + 2.81i)14-s + (0.173 + 0.984i)16-s + (−2.08 + 5.73i)17-s + 1.22i·18-s + (−2.69 − 3.42i)19-s + ⋯
L(s)  = 1  + (0.241 − 0.664i)2-s + (0.757 − 0.133i)3-s + (−0.383 − 0.321i)4-s + (0.0944 − 0.535i)6-s + (−1.43 − 0.826i)7-s + (−0.306 + 0.176i)8-s + (−0.384 + 0.139i)9-s + (0.137 + 0.238i)11-s + (−0.332 − 0.192i)12-s + (−1.10 − 0.194i)13-s + (−0.895 + 0.751i)14-s + (0.0434 + 0.246i)16-s + (−0.506 + 1.39i)17-s + 0.289i·18-s + (−0.618 − 0.785i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.793 - 0.608i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.793 - 0.608i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $-0.793 - 0.608i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (99, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ -0.793 - 0.608i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.138335 + 0.407987i\)
\(L(\frac12)\) \(\approx\) \(0.138335 + 0.407987i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.342 + 0.939i)T \)
5 \( 1 \)
19 \( 1 + (2.69 + 3.42i)T \)
good3 \( 1 + (-1.31 + 0.231i)T + (2.81 - 1.02i)T^{2} \)
7 \( 1 + (3.78 + 2.18i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.457 - 0.792i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (3.97 + 0.700i)T + (12.2 + 4.44i)T^{2} \)
17 \( 1 + (2.08 - 5.73i)T + (-13.0 - 10.9i)T^{2} \)
23 \( 1 + (-0.567 + 0.676i)T + (-3.99 - 22.6i)T^{2} \)
29 \( 1 + (-8.44 + 3.07i)T + (22.2 - 18.6i)T^{2} \)
31 \( 1 + (3.11 - 5.39i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 10.2iT - 37T^{2} \)
41 \( 1 + (-0.760 - 4.31i)T + (-38.5 + 14.0i)T^{2} \)
43 \( 1 + (4.60 + 5.48i)T + (-7.46 + 42.3i)T^{2} \)
47 \( 1 + (0.572 + 1.57i)T + (-36.0 + 30.2i)T^{2} \)
53 \( 1 + (-0.335 + 0.399i)T + (-9.20 - 52.1i)T^{2} \)
59 \( 1 + (2.68 + 0.976i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (7.92 + 6.64i)T + (10.5 + 60.0i)T^{2} \)
67 \( 1 + (5.21 + 14.3i)T + (-51.3 + 43.0i)T^{2} \)
71 \( 1 + (-4.16 + 3.49i)T + (12.3 - 69.9i)T^{2} \)
73 \( 1 + (-5.49 + 0.968i)T + (68.5 - 24.9i)T^{2} \)
79 \( 1 + (0.417 + 2.36i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (2.26 + 1.30i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (0.171 - 0.974i)T + (-83.6 - 30.4i)T^{2} \)
97 \( 1 + (2.35 - 6.47i)T + (-74.3 - 62.3i)T^{2} \)
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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

−9.554251232853488434765919584274, −8.924492005446455530894727553432, −7.972805988246604540127148877060, −6.93255083135686820295723750680, −6.20677951225267042913320252010, −4.84850485593423597561444391447, −3.83963918859021461140050227198, −3.02439087591215374757225134967, −2.10174069832883503824917036701, −0.15257514101815035125898843450, 2.62532915217039127510140916881, 3.10132772859069907795307090294, 4.36179784719470097523869208302, 5.47736169104322058627607160008, 6.35269335067636665255454317765, 7.03274461434901978450587411929, 8.137122551843993986617115116417, 8.901329147583777816172561496343, 9.480994879628741334177548623466, 10.09180018264839005194045525633

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