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Properties

Label 2-950-95.24-c1-0-4
Degree $2$
Conductor $950$
Sign $0.734 - 0.678i$
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 + (0.0355 + 0.00627i)3-s + (−0.766 + 0.642i)4-s + (−0.00627 − 0.0355i)6-s + (−1.59 + 0.918i)7-s + (0.866 + 0.500i)8-s + (−2.81 − 1.02i)9-s + (1.23 − 2.13i)11-s + (−0.0313 + 0.0180i)12-s + (−2.35 + 0.415i)13-s + (1.40 + 1.18i)14-s + (0.173 − 0.984i)16-s + (2.30 + 6.33i)17-s + 2.99i·18-s + (4.34 − 0.298i)19-s + ⋯
L(s)  = 1  + (−0.241 − 0.664i)2-s + (0.0205 + 0.00362i)3-s + (−0.383 + 0.321i)4-s + (−0.00256 − 0.0145i)6-s + (−0.601 + 0.347i)7-s + (0.306 + 0.176i)8-s + (−0.939 − 0.341i)9-s + (0.371 − 0.643i)11-s + (−0.00903 + 0.00521i)12-s + (−0.653 + 0.115i)13-s + (0.376 + 0.315i)14-s + (0.0434 − 0.246i)16-s + (0.559 + 1.53i)17-s + 0.706i·18-s + (0.997 − 0.0684i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.734 - 0.678i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.734 - 0.678i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $0.734 - 0.678i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (499, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ 0.734 - 0.678i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.814257 + 0.318472i\)
\(L(\frac12)\) \(\approx\) \(0.814257 + 0.318472i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-4.34 + 0.298i)T \)
good3 \( 1 + (-0.0355 - 0.00627i)T + (2.81 + 1.02i)T^{2} \)
7 \( 1 + (1.59 - 0.918i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.23 + 2.13i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.35 - 0.415i)T + (12.2 - 4.44i)T^{2} \)
17 \( 1 + (-2.30 - 6.33i)T + (-13.0 + 10.9i)T^{2} \)
23 \( 1 + (-1.04 - 1.24i)T + (-3.99 + 22.6i)T^{2} \)
29 \( 1 + (-3.10 - 1.12i)T + (22.2 + 18.6i)T^{2} \)
31 \( 1 + (-1.75 - 3.03i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 6.00iT - 37T^{2} \)
41 \( 1 + (1.38 - 7.85i)T + (-38.5 - 14.0i)T^{2} \)
43 \( 1 + (3.67 - 4.37i)T + (-7.46 - 42.3i)T^{2} \)
47 \( 1 + (4.27 - 11.7i)T + (-36.0 - 30.2i)T^{2} \)
53 \( 1 + (1.23 + 1.47i)T + (-9.20 + 52.1i)T^{2} \)
59 \( 1 + (4.46 - 1.62i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (-10.4 + 8.78i)T + (10.5 - 60.0i)T^{2} \)
67 \( 1 + (-1.27 + 3.49i)T + (-51.3 - 43.0i)T^{2} \)
71 \( 1 + (-4.46 - 3.74i)T + (12.3 + 69.9i)T^{2} \)
73 \( 1 + (0.502 + 0.0886i)T + (68.5 + 24.9i)T^{2} \)
79 \( 1 + (2.10 - 11.9i)T + (-74.2 - 27.0i)T^{2} \)
83 \( 1 + (-5.39 + 3.11i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (2.95 + 16.7i)T + (-83.6 + 30.4i)T^{2} \)
97 \( 1 + (4.67 + 12.8i)T + (-74.3 + 62.3i)T^{2} \)
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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

−9.958572565461778772141923706749, −9.500878695017411464570322136866, −8.524740865035863647584980326177, −7.981733815357389080384819357507, −6.59043327347464481267435228962, −5.88572744331417801135327997995, −4.79375769061868564460791505066, −3.38628803843535877620970315699, −2.94135223391331732136993564023, −1.29620428235868336115221259941, 0.48109637766238436238702868730, 2.45374695518609719978614858681, 3.61493891805187483909932162149, 4.97097755702456171864122510917, 5.52982236794965006634471127915, 6.78702219380990186184122376540, 7.28551214035928360157516226244, 8.170503518007567868391340407342, 9.188706347608589882367327873933, 9.751405640078550939749345964756

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