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Properties

Label 2-950-19.9-c1-0-11
Degree $2$
Conductor $950$
Sign $0.827 + 0.560i$
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  + (−0.173 − 0.984i)2-s + (−2.49 + 2.09i)3-s + (−0.939 + 0.342i)4-s + (2.49 + 2.09i)6-s + (−0.556 + 0.964i)7-s + (0.5 + 0.866i)8-s + (1.31 − 7.46i)9-s + (0.0761 + 0.131i)11-s + (1.62 − 2.81i)12-s + (−2.66 − 2.23i)13-s + (1.04 + 0.380i)14-s + (0.766 − 0.642i)16-s + (−0.0290 − 0.164i)17-s − 7.58·18-s + (−3.08 + 3.08i)19-s + ⋯
L(s)  = 1  + (−0.122 − 0.696i)2-s + (−1.43 + 1.20i)3-s + (−0.469 + 0.171i)4-s + (1.01 + 0.853i)6-s + (−0.210 + 0.364i)7-s + (0.176 + 0.306i)8-s + (0.438 − 2.48i)9-s + (0.0229 + 0.0397i)11-s + (0.469 − 0.813i)12-s + (−0.738 − 0.619i)13-s + (0.279 + 0.101i)14-s + (0.191 − 0.160i)16-s + (−0.00704 − 0.0399i)17-s − 1.78·18-s + (−0.707 + 0.706i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.827 + 0.560i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.827 + 0.560i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $0.827 + 0.560i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (351, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ 0.827 + 0.560i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.555349 - 0.170353i\)
\(L(\frac12)\) \(\approx\) \(0.555349 - 0.170353i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.173 + 0.984i)T \)
5 \( 1 \)
19 \( 1 + (3.08 - 3.08i)T \)
good3 \( 1 + (2.49 - 2.09i)T + (0.520 - 2.95i)T^{2} \)
7 \( 1 + (0.556 - 0.964i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.0761 - 0.131i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.66 + 2.23i)T + (2.25 + 12.8i)T^{2} \)
17 \( 1 + (0.0290 + 0.164i)T + (-15.9 + 5.81i)T^{2} \)
23 \( 1 + (5.83 - 2.12i)T + (17.6 - 14.7i)T^{2} \)
29 \( 1 + (-0.881 + 4.99i)T + (-27.2 - 9.91i)T^{2} \)
31 \( 1 + (-4.34 + 7.52i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 3.71T + 37T^{2} \)
41 \( 1 + (-3.70 + 3.10i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 + (-9.50 - 3.46i)T + (32.9 + 27.6i)T^{2} \)
47 \( 1 + (-1.46 + 8.28i)T + (-44.1 - 16.0i)T^{2} \)
53 \( 1 + (-13.3 + 4.87i)T + (40.6 - 34.0i)T^{2} \)
59 \( 1 + (-2.35 - 13.3i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (4.92 - 1.79i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (0.450 - 2.55i)T + (-62.9 - 22.9i)T^{2} \)
71 \( 1 + (-1.70 - 0.622i)T + (54.3 + 45.6i)T^{2} \)
73 \( 1 + (7.79 - 6.53i)T + (12.6 - 71.8i)T^{2} \)
79 \( 1 + (-4.00 + 3.35i)T + (13.7 - 77.7i)T^{2} \)
83 \( 1 + (-3.03 + 5.26i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-9.18 - 7.70i)T + (15.4 + 87.6i)T^{2} \)
97 \( 1 + (2.16 + 12.2i)T + (-91.1 + 33.1i)T^{2} \)
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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

−9.992982612921409259621760113935, −9.698922242309059536786023723761, −8.614734048088264019724475991149, −7.43432327631369835186111649017, −5.89880929404473818708041076247, −5.77368874744885460578380306414, −4.42005767122416165739518964943, −3.97687800717276370743322944286, −2.51580766938782178752756151374, −0.50308066097962948220459957093, 0.826761041435632227534271036474, 2.26275377752087840100650435422, 4.34410095593624040410270920905, 5.10696251942043680621200699956, 6.12807932201931876262798181878, 6.66511695604079023160904442692, 7.31142971488573842632023435177, 8.082148741546082752324135608643, 9.178874480212137627293251311406, 10.37424466621767949227428024542

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