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Properties

Label 2-950-19.7-c1-0-11
Degree $2$
Conductor $950$
Sign $0.988 - 0.149i$
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  + (−0.5 − 0.866i)2-s + (0.851 + 1.47i)3-s + (−0.499 + 0.866i)4-s + (0.851 − 1.47i)6-s − 3.74·7-s + 0.999·8-s + (0.0492 − 0.0852i)9-s + 3.64·11-s − 1.70·12-s + (3.01 − 5.23i)13-s + (1.87 + 3.24i)14-s + (−0.5 − 0.866i)16-s + (2.04 + 3.54i)17-s − 0.0984·18-s + (−0.697 + 4.30i)19-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (0.491 + 0.851i)3-s + (−0.249 + 0.433i)4-s + (0.347 − 0.602i)6-s − 1.41·7-s + 0.353·8-s + (0.0164 − 0.0284i)9-s + 1.09·11-s − 0.491·12-s + (0.837 − 1.45i)13-s + (0.500 + 0.866i)14-s + (−0.125 − 0.216i)16-s + (0.497 + 0.860i)17-s − 0.0231·18-s + (−0.160 + 0.987i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.988 - 0.149i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.988 - 0.149i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $0.988 - 0.149i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (501, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ 0.988 - 0.149i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.45586 + 0.109422i\)
\(L(\frac12)\) \(\approx\) \(1.45586 + 0.109422i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 + 0.866i)T \)
5 \( 1 \)
19 \( 1 + (0.697 - 4.30i)T \)
good3 \( 1 + (-0.851 - 1.47i)T + (-1.5 + 2.59i)T^{2} \)
7 \( 1 + 3.74T + 7T^{2} \)
11 \( 1 - 3.64T + 11T^{2} \)
13 \( 1 + (-3.01 + 5.23i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2.04 - 3.54i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (2.34 - 4.05i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.32 + 5.75i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 10.8T + 31T^{2} \)
37 \( 1 + 7.75T + 37T^{2} \)
41 \( 1 + (-3.99 - 6.92i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.19 - 3.80i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (0.871 - 1.50i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.871 + 1.50i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.67 - 6.36i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.15 - 1.99i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.37 + 5.83i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-0.994 - 1.72i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-4.59 - 7.95i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (3.07 + 5.33i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 5.69T + 83T^{2} \)
89 \( 1 + (-5.53 + 9.59i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-0.752 - 1.30i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.09476050460318343066311943802, −9.502037834537735629897810408046, −8.575157624495770154590223007805, −7.948620311895090965164537512826, −6.47593018299442878523527914268, −5.89627015045852875932512408805, −4.22387095710831998726944779110, −3.56381278594429813929473774706, −2.98338195675473807541413315517, −1.10986719782811446732585599109, 0.968694144587783523765367278157, 2.37259887063712943202966284659, 3.63674442117526412307552937361, 4.77318988284818362236752083793, 6.31471488408747991243176621673, 6.69953845689724100296393966062, 7.19805814710681092704666911934, 8.527222722819851380197993327648, 8.952846254827565920602837057112, 9.724639948109676890368852014647

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