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Properties

Label 2-950-25.21-c1-0-9
Degree $2$
Conductor $950$
Sign $0.999 - 0.0361i$
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  + (0.309 − 0.951i)2-s + (−1.83 − 1.33i)3-s + (−0.809 − 0.587i)4-s + (−1.65 − 1.50i)5-s + (−1.83 + 1.33i)6-s + 4.52·7-s + (−0.809 + 0.587i)8-s + (0.665 + 2.04i)9-s + (−1.93 + 1.11i)10-s + (−1.44 + 4.43i)11-s + (0.701 + 2.15i)12-s + (1.69 + 5.22i)13-s + (1.39 − 4.29i)14-s + (1.04 + 4.96i)15-s + (0.309 + 0.951i)16-s + (−5.72 + 4.15i)17-s + ⋯
L(s)  = 1  + (0.218 − 0.672i)2-s + (−1.06 − 0.770i)3-s + (−0.404 − 0.293i)4-s + (−0.741 − 0.671i)5-s + (−0.749 + 0.544i)6-s + 1.70·7-s + (−0.286 + 0.207i)8-s + (0.221 + 0.682i)9-s + (−0.613 + 0.351i)10-s + (−0.434 + 1.33i)11-s + (0.202 + 0.623i)12-s + (0.470 + 1.44i)13-s + (0.373 − 1.14i)14-s + (0.268 + 1.28i)15-s + (0.0772 + 0.237i)16-s + (−1.38 + 1.00i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0361i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0361i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $0.999 - 0.0361i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (571, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ 0.999 - 0.0361i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.804614 + 0.0145572i\)
\(L(\frac12)\) \(\approx\) \(0.804614 + 0.0145572i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.309 + 0.951i)T \)
5 \( 1 + (1.65 + 1.50i)T \)
19 \( 1 + (-0.809 + 0.587i)T \)
good3 \( 1 + (1.83 + 1.33i)T + (0.927 + 2.85i)T^{2} \)
7 \( 1 - 4.52T + 7T^{2} \)
11 \( 1 + (1.44 - 4.43i)T + (-8.89 - 6.46i)T^{2} \)
13 \( 1 + (-1.69 - 5.22i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (5.72 - 4.15i)T + (5.25 - 16.1i)T^{2} \)
23 \( 1 + (-0.440 + 1.35i)T + (-18.6 - 13.5i)T^{2} \)
29 \( 1 + (-3.33 - 2.42i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (3.16 - 2.30i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-0.259 - 0.799i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (1.04 + 3.22i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 2.84T + 43T^{2} \)
47 \( 1 + (-1.88 - 1.37i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (4.07 + 2.96i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (2.21 + 6.80i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (2.35 - 7.24i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (11.6 - 8.50i)T + (20.7 - 63.7i)T^{2} \)
71 \( 1 + (-9.32 - 6.77i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (0.582 - 1.79i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-0.0773 - 0.0562i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (10.7 - 7.82i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + (-4.69 + 14.4i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (10.5 + 7.68i)T + (29.9 + 92.2i)T^{2} \)
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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

−10.49268006235041123384121360926, −9.049362837443410804672890686683, −8.464748805737052076950791689804, −7.42248585693189411413140704406, −6.70258251105369848431532252078, −5.43688765330726083492654823146, −4.59684669600514059936726875029, −4.22684043142755200301865207699, −1.96930844106556434440002529019, −1.37501976249684364421703674932, 0.43679159887056817966443121287, 2.89618065112181558814363255971, 4.11941975937581612707488456705, 4.92273876632756109217465235085, 5.55993841744579930731675197900, 6.39273489623233134766959499162, 7.72906704758173398154110341210, 8.028083601140706649410930471527, 9.015127174475888769861389348693, 10.48316390439210140370038234833

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