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Properties

Label 2-950-19.5-c1-0-22
Degree $2$
Conductor $950$
Sign $0.324 + 0.945i$
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 + (−0.0989 + 0.561i)3-s + (0.766 − 0.642i)4-s + (0.0989 + 0.561i)6-s + (−2.10 − 3.64i)7-s + (0.500 − 0.866i)8-s + (2.51 + 0.914i)9-s + (−1.66 + 2.87i)11-s + (0.284 + 0.493i)12-s + (−0.485 − 2.75i)13-s + (−3.21 − 2.70i)14-s + (0.173 − 0.984i)16-s + (4.32 − 1.57i)17-s + 2.67·18-s + (−0.508 − 4.32i)19-s + ⋯
L(s)  = 1  + (0.664 − 0.241i)2-s + (−0.0571 + 0.324i)3-s + (0.383 − 0.321i)4-s + (0.0404 + 0.229i)6-s + (−0.794 − 1.37i)7-s + (0.176 − 0.306i)8-s + (0.837 + 0.304i)9-s + (−0.501 + 0.868i)11-s + (0.0822 + 0.142i)12-s + (−0.134 − 0.763i)13-s + (−0.860 − 0.722i)14-s + (0.0434 − 0.246i)16-s + (1.04 − 0.381i)17-s + 0.630·18-s + (−0.116 − 0.993i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.324 + 0.945i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.324 + 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $0.324 + 0.945i$
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: \((2,\ 950,\ (\ :1/2),\ 0.324 + 0.945i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.73011 - 1.23554i\)
\(L(\frac12)\) \(\approx\) \(1.73011 - 1.23554i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.508 + 4.32i)T \)
good3 \( 1 + (0.0989 - 0.561i)T + (-2.81 - 1.02i)T^{2} \)
7 \( 1 + (2.10 + 3.64i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.66 - 2.87i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.485 + 2.75i)T + (-12.2 + 4.44i)T^{2} \)
17 \( 1 + (-4.32 + 1.57i)T + (13.0 - 10.9i)T^{2} \)
23 \( 1 + (-6.48 + 5.43i)T + (3.99 - 22.6i)T^{2} \)
29 \( 1 + (0.414 + 0.150i)T + (22.2 + 18.6i)T^{2} \)
31 \( 1 + (4.93 + 8.54i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 4.04T + 37T^{2} \)
41 \( 1 + (1.90 - 10.8i)T + (-38.5 - 14.0i)T^{2} \)
43 \( 1 + (-1.03 - 0.869i)T + (7.46 + 42.3i)T^{2} \)
47 \( 1 + (5.39 + 1.96i)T + (36.0 + 30.2i)T^{2} \)
53 \( 1 + (-0.586 + 0.492i)T + (9.20 - 52.1i)T^{2} \)
59 \( 1 + (-5.17 + 1.88i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (-5.05 + 4.23i)T + (10.5 - 60.0i)T^{2} \)
67 \( 1 + (3.77 + 1.37i)T + (51.3 + 43.0i)T^{2} \)
71 \( 1 + (-7.41 - 6.22i)T + (12.3 + 69.9i)T^{2} \)
73 \( 1 + (-0.578 + 3.28i)T + (-68.5 - 24.9i)T^{2} \)
79 \( 1 + (0.964 - 5.46i)T + (-74.2 - 27.0i)T^{2} \)
83 \( 1 + (-3.75 - 6.50i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (1.91 + 10.8i)T + (-83.6 + 30.4i)T^{2} \)
97 \( 1 + (13.3 - 4.86i)T + (74.3 - 62.3i)T^{2} \)
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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

−9.920887793494494418224323227996, −9.587496433090666886073393119116, −7.87148227175083406566332280977, −7.22690066073653749683989239969, −6.55623126487447981164450644288, −5.17252904973297345090220134735, −4.55248796414113089374178534820, −3.62477237342983145945220569512, −2.59675779976186435175767549085, −0.847603573498386047926703189941, 1.66743676794697216523672545831, 3.02960314142778573593306278346, 3.76415510748850860846608720074, 5.28866264180436228730218014531, 5.77159669237289792159576639085, 6.69150578176245059387400108129, 7.46320466042008588978836358728, 8.538777162599261784650205925068, 9.285350983723273118788239696567, 10.17214636804161275441521448053

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