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Properties

Label 2-950-19.16-c1-0-4
Degree $2$
Conductor $950$
Sign $-0.416 - 0.908i$
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  + (−0.766 − 0.642i)2-s + (−2.57 + 0.936i)3-s + (0.173 + 0.984i)4-s + (2.57 + 0.936i)6-s + (1.92 + 3.33i)7-s + (0.500 − 0.866i)8-s + (3.44 − 2.89i)9-s + (−2.86 + 4.96i)11-s + (−1.36 − 2.37i)12-s + (5.05 + 1.84i)13-s + (0.668 − 3.79i)14-s + (−0.939 + 0.342i)16-s + (−1.05 − 0.883i)17-s − 4.50·18-s + (4.23 − 1.03i)19-s + ⋯
L(s)  = 1  + (−0.541 − 0.454i)2-s + (−1.48 + 0.540i)3-s + (0.0868 + 0.492i)4-s + (1.05 + 0.382i)6-s + (0.727 + 1.25i)7-s + (0.176 − 0.306i)8-s + (1.14 − 0.964i)9-s + (−0.863 + 1.49i)11-s + (−0.395 − 0.684i)12-s + (1.40 + 0.510i)13-s + (0.178 − 1.01i)14-s + (−0.234 + 0.0855i)16-s + (−0.255 − 0.214i)17-s − 1.06·18-s + (0.971 − 0.238i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.416 - 0.908i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.416 - 0.908i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $-0.416 - 0.908i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (301, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ -0.416 - 0.908i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.366360 + 0.571073i\)
\(L(\frac12)\) \(\approx\) \(0.366360 + 0.571073i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.766 + 0.642i)T \)
5 \( 1 \)
19 \( 1 + (-4.23 + 1.03i)T \)
good3 \( 1 + (2.57 - 0.936i)T + (2.29 - 1.92i)T^{2} \)
7 \( 1 + (-1.92 - 3.33i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.86 - 4.96i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-5.05 - 1.84i)T + (9.95 + 8.35i)T^{2} \)
17 \( 1 + (1.05 + 0.883i)T + (2.95 + 16.7i)T^{2} \)
23 \( 1 + (0.274 + 1.55i)T + (-21.6 + 7.86i)T^{2} \)
29 \( 1 + (-5.22 + 4.38i)T + (5.03 - 28.5i)T^{2} \)
31 \( 1 + (-3.07 - 5.32i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 1.89T + 37T^{2} \)
41 \( 1 + (3.39 - 1.23i)T + (31.4 - 26.3i)T^{2} \)
43 \( 1 + (0.568 - 3.22i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (5.55 - 4.65i)T + (8.16 - 46.2i)T^{2} \)
53 \( 1 + (-2.04 - 11.5i)T + (-49.8 + 18.1i)T^{2} \)
59 \( 1 + (2.00 + 1.68i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (1.05 + 5.99i)T + (-57.3 + 20.8i)T^{2} \)
67 \( 1 + (-4.81 + 4.03i)T + (11.6 - 65.9i)T^{2} \)
71 \( 1 + (2.16 - 12.2i)T + (-66.7 - 24.2i)T^{2} \)
73 \( 1 + (8.85 - 3.22i)T + (55.9 - 46.9i)T^{2} \)
79 \( 1 + (7.27 - 2.64i)T + (60.5 - 50.7i)T^{2} \)
83 \( 1 + (2.24 + 3.88i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-0.964 - 0.351i)T + (68.1 + 57.2i)T^{2} \)
97 \( 1 + (-1.01 - 0.849i)T + (16.8 + 95.5i)T^{2} \)
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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

−10.35671028826583303054111273976, −9.717319067508756674500681158338, −8.807841392511698399705376387518, −7.942914022464948954003380233982, −6.77693787526143459493372128931, −5.91381621830371472486449278963, −4.96536692072093558782562456418, −4.42897596094909939594097052511, −2.71601176450414233205246191875, −1.40708933745624829187890853746, 0.56258614690692538389967935084, 1.30263400029351297438197087780, 3.49174160413966746804266217390, 4.88375829115804968883644316414, 5.66576636886502171156166277362, 6.29117698950148178265135721651, 7.21060996973510155078899003540, 7.993853869844903722068176713241, 8.600812575851380282113338466636, 10.23993230147926698266981072148

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