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Properties

Label 2-950-19.5-c1-0-18
Degree $2$
Conductor $950$
Sign $0.703 + 0.710i$
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  + (−0.939 + 0.342i)2-s + (−0.111 + 0.634i)3-s + (0.766 − 0.642i)4-s + (−0.111 − 0.634i)6-s + (−0.213 − 0.369i)7-s + (−0.500 + 0.866i)8-s + (2.42 + 0.883i)9-s + (1.50 − 2.60i)11-s + (0.322 + 0.558i)12-s + (−0.831 − 4.71i)13-s + (0.327 + 0.274i)14-s + (0.173 − 0.984i)16-s + (−4.46 + 1.62i)17-s − 2.58·18-s + (−2.34 − 3.67i)19-s + ⋯
L(s)  = 1  + (−0.664 + 0.241i)2-s + (−0.0646 + 0.366i)3-s + (0.383 − 0.321i)4-s + (−0.0457 − 0.259i)6-s + (−0.0807 − 0.139i)7-s + (−0.176 + 0.306i)8-s + (0.809 + 0.294i)9-s + (0.453 − 0.786i)11-s + (0.0930 + 0.161i)12-s + (−0.230 − 1.30i)13-s + (0.0874 + 0.0733i)14-s + (0.0434 − 0.246i)16-s + (−1.08 + 0.394i)17-s − 0.609·18-s + (−0.537 − 0.843i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.703 + 0.710i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.703 + 0.710i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $0.703 + 0.710i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (651, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ 0.703 + 0.710i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.927225 - 0.386461i\)
\(L(\frac12)\) \(\approx\) \(0.927225 - 0.386461i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.939 - 0.342i)T \)
5 \( 1 \)
19 \( 1 + (2.34 + 3.67i)T \)
good3 \( 1 + (0.111 - 0.634i)T + (-2.81 - 1.02i)T^{2} \)
7 \( 1 + (0.213 + 0.369i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.50 + 2.60i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.831 + 4.71i)T + (-12.2 + 4.44i)T^{2} \)
17 \( 1 + (4.46 - 1.62i)T + (13.0 - 10.9i)T^{2} \)
23 \( 1 + (-4.00 + 3.36i)T + (3.99 - 22.6i)T^{2} \)
29 \( 1 + (-0.536 - 0.195i)T + (22.2 + 18.6i)T^{2} \)
31 \( 1 + (-0.113 - 0.196i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 9.76T + 37T^{2} \)
41 \( 1 + (-1.60 + 9.12i)T + (-38.5 - 14.0i)T^{2} \)
43 \( 1 + (-0.777 - 0.652i)T + (7.46 + 42.3i)T^{2} \)
47 \( 1 + (-6.78 - 2.46i)T + (36.0 + 30.2i)T^{2} \)
53 \( 1 + (-8.29 + 6.95i)T + (9.20 - 52.1i)T^{2} \)
59 \( 1 + (0.330 - 0.120i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (-0.0978 + 0.0820i)T + (10.5 - 60.0i)T^{2} \)
67 \( 1 + (0.246 + 0.0895i)T + (51.3 + 43.0i)T^{2} \)
71 \( 1 + (-0.343 - 0.287i)T + (12.3 + 69.9i)T^{2} \)
73 \( 1 + (-1.48 + 8.41i)T + (-68.5 - 24.9i)T^{2} \)
79 \( 1 + (2.31 - 13.1i)T + (-74.2 - 27.0i)T^{2} \)
83 \( 1 + (-0.550 - 0.954i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (1.75 + 9.98i)T + (-83.6 + 30.4i)T^{2} \)
97 \( 1 + (-14.5 + 5.28i)T + (74.3 - 62.3i)T^{2} \)
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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

−10.05276033273625261475983930406, −8.870565226896181686698146420833, −8.577390235897628900722018075488, −7.32420645563700925757301475255, −6.75461277047328382633974316023, −5.65464968463626699959217748834, −4.71503557504593888660196617220, −3.58261686574866662498762586039, −2.25525731580797020062710085470, −0.62406597124482677650944239204, 1.40886526491134789241591481805, 2.29356017783613995092193000028, 3.86856855346292612138619164331, 4.67131281103092724204538576758, 6.20609762205194851884396904857, 6.98437339570511922602698164813, 7.41738767058379115740662988657, 8.739141222272414550036706132475, 9.281885214936967957857901242637, 10.00477776898926018275980200698

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