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Properties

Label 2-950-95.48-c1-0-4
Degree $2$
Conductor $950$
Sign $0.200 - 0.979i$
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  + (0.573 − 0.819i)2-s + (−0.190 + 2.17i)3-s + (−0.342 − 0.939i)4-s + (1.67 + 1.40i)6-s + (−0.115 − 0.431i)7-s + (−0.965 − 0.258i)8-s + (−1.74 − 0.307i)9-s + (2.37 + 4.12i)11-s + (2.11 − 0.565i)12-s + (−2.40 + 0.210i)13-s + (−0.419 − 0.152i)14-s + (−0.766 + 0.642i)16-s + (−0.560 − 0.392i)17-s + (−1.25 + 1.25i)18-s + (2.15 + 3.79i)19-s + ⋯
L(s)  = 1  + (0.405 − 0.579i)2-s + (−0.109 + 1.25i)3-s + (−0.171 − 0.469i)4-s + (0.683 + 0.573i)6-s + (−0.0436 − 0.163i)7-s + (−0.341 − 0.0915i)8-s + (−0.582 − 0.102i)9-s + (0.717 + 1.24i)11-s + (0.609 − 0.163i)12-s + (−0.666 + 0.0582i)13-s + (−0.112 − 0.0408i)14-s + (−0.191 + 0.160i)16-s + (−0.135 − 0.0951i)17-s + (−0.295 + 0.295i)18-s + (0.493 + 0.869i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.200 - 0.979i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.200 - 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $0.200 - 0.979i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (143, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ 0.200 - 0.979i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.27951 + 1.04390i\)
\(L(\frac12)\) \(\approx\) \(1.27951 + 1.04390i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.573 + 0.819i)T \)
5 \( 1 \)
19 \( 1 + (-2.15 - 3.79i)T \)
good3 \( 1 + (0.190 - 2.17i)T + (-2.95 - 0.520i)T^{2} \)
7 \( 1 + (0.115 + 0.431i)T + (-6.06 + 3.5i)T^{2} \)
11 \( 1 + (-2.37 - 4.12i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.40 - 0.210i)T + (12.8 - 2.25i)T^{2} \)
17 \( 1 + (0.560 + 0.392i)T + (5.81 + 15.9i)T^{2} \)
23 \( 1 + (0.404 + 0.188i)T + (14.7 + 17.6i)T^{2} \)
29 \( 1 + (0.892 - 5.06i)T + (-27.2 - 9.91i)T^{2} \)
31 \( 1 + (-2.27 - 1.31i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.48 + 1.48i)T + 37iT^{2} \)
41 \( 1 + (0.333 + 0.397i)T + (-7.11 + 40.3i)T^{2} \)
43 \( 1 + (-3.94 - 8.46i)T + (-27.6 + 32.9i)T^{2} \)
47 \( 1 + (3.02 + 4.32i)T + (-16.0 + 44.1i)T^{2} \)
53 \( 1 + (3.42 - 7.35i)T + (-34.0 - 40.6i)T^{2} \)
59 \( 1 + (-0.0555 - 0.315i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (7.77 - 2.82i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (9.20 - 6.44i)T + (22.9 - 62.9i)T^{2} \)
71 \( 1 + (-0.879 + 2.41i)T + (-54.3 - 45.6i)T^{2} \)
73 \( 1 + (-4.32 - 0.378i)T + (71.8 + 12.6i)T^{2} \)
79 \( 1 + (-12.3 + 10.3i)T + (13.7 - 77.7i)T^{2} \)
83 \( 1 + (0.838 - 0.224i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 + (-12.7 - 10.7i)T + (15.4 + 87.6i)T^{2} \)
97 \( 1 + (-1.72 + 2.46i)T + (-33.1 - 91.1i)T^{2} \)
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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

−10.28496113895015571671958297030, −9.538124976789429647690270597553, −9.106705578237321261988825739132, −7.67192419645988827752567103998, −6.72838797448316061420469378588, −5.51520712911645292962289928720, −4.66492730217243807445138994959, −4.10824326331642104670188119640, −3.10651927710579851224349221222, −1.67120059088464352506741871933, 0.70631728226833540334394327820, 2.27876823591755395804716949540, 3.45731873513282648467244040251, 4.71813259983621368215084651508, 5.86040107993777583585285099186, 6.41363025397264292592119193956, 7.24011733113983145621592573446, 7.927526246834797866181192799493, 8.780624508154398867323350644993, 9.619095700562876739351186352144

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