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Properties

Label 2-950-19.11-c1-0-16
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: $\chi_{950} (201, \cdot )$
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

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

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