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Properties

Label 2-950-25.21-c1-0-19
Degree $2$
Conductor $950$
Sign $-0.794 - 0.606i$
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 + (2.31 + 1.68i)3-s + (−0.809 − 0.587i)4-s + (1.05 + 1.97i)5-s + (−2.31 + 1.68i)6-s − 0.662·7-s + (0.809 − 0.587i)8-s + (1.61 + 4.95i)9-s + (−2.20 + 0.393i)10-s + (0.359 − 1.10i)11-s + (−0.885 − 2.72i)12-s + (0.939 + 2.89i)13-s + (0.204 − 0.629i)14-s + (−0.877 + 6.34i)15-s + (0.309 + 0.951i)16-s + (3.38 − 2.45i)17-s + ⋯
L(s)  = 1  + (−0.218 + 0.672i)2-s + (1.33 + 0.972i)3-s + (−0.404 − 0.293i)4-s + (0.471 + 0.881i)5-s + (−0.946 + 0.687i)6-s − 0.250·7-s + (0.286 − 0.207i)8-s + (0.536 + 1.65i)9-s + (−0.696 + 0.124i)10-s + (0.108 − 0.333i)11-s + (−0.255 − 0.786i)12-s + (0.260 + 0.801i)13-s + (0.0546 − 0.168i)14-s + (−0.226 + 1.63i)15-s + (0.0772 + 0.237i)16-s + (0.821 − 0.596i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.794 - 0.606i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.794 - 0.606i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $-0.794 - 0.606i$
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: \((2,\ 950,\ (\ :1/2),\ -0.794 - 0.606i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.727001 + 2.15104i\)
\(L(\frac12)\) \(\approx\) \(0.727001 + 2.15104i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.05 - 1.97i)T \)
19 \( 1 + (-0.809 + 0.587i)T \)
good3 \( 1 + (-2.31 - 1.68i)T + (0.927 + 2.85i)T^{2} \)
7 \( 1 + 0.662T + 7T^{2} \)
11 \( 1 + (-0.359 + 1.10i)T + (-8.89 - 6.46i)T^{2} \)
13 \( 1 + (-0.939 - 2.89i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (-3.38 + 2.45i)T + (5.25 - 16.1i)T^{2} \)
23 \( 1 + (1.45 - 4.48i)T + (-18.6 - 13.5i)T^{2} \)
29 \( 1 + (4.90 + 3.56i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-0.651 + 0.473i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-0.0115 - 0.0355i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (2.03 + 6.25i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 1.95T + 43T^{2} \)
47 \( 1 + (2.98 + 2.16i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (-2.56 - 1.86i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-3.89 - 11.9i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-3.54 + 10.9i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (-8.95 + 6.50i)T + (20.7 - 63.7i)T^{2} \)
71 \( 1 + (0.360 + 0.261i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-3.26 + 10.0i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (4.89 + 3.55i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-0.338 + 0.245i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + (2.67 - 8.24i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (-5.22 - 3.79i)T + (29.9 + 92.2i)T^{2} \)
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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

−9.881609645113313695248013901447, −9.559330040731394895885193725856, −8.859618572708843406347994538847, −7.87754948633416411670185599940, −7.19753807214161027668892910353, −6.14813351403475370059645238821, −5.17029830264986942061747516261, −3.89134035090245910918614638005, −3.26558279911325839875741577854, −2.05202152619968615137624706129, 1.05488951615940359804825904356, 1.96253782599721495500270219432, 3.00777925923245354614656604253, 3.93807221561149871250434441294, 5.28610491389937310750045404269, 6.41780114042853668746133062984, 7.54803597606063047521039727938, 8.290990414949590412309334791728, 8.714390627143787552521139142161, 9.692449397760124478666264153187

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