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Properties

Label 2-950-19.5-c1-0-3
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

\(L(1)\) \(\approx\) \(0.600700 + 0.841148i\)
\(L(\frac12)\) \(\approx\) \(0.600700 + 0.841148i\)
\(L(\frac{3}{2})\) 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

−10.04249152001698812309303413778, −9.394939532021801858313229398164, −8.585708009391516772210355836341, −7.83121497362250295704399522886, −7.08855344577917825446225797428, −6.14800726553656418300235308650, −5.14633487483827238864266063332, −4.25130723507445761539514032277, −2.21724342542297153254726731561, −1.88423876931040130808134872999, 0.58158460438328266589357179642, 1.90747606433232914489852357307, 3.51056159342988687833523820398, 4.19875772421940157209429856494, 5.34655810737555313433326236019, 6.65568358492779907805527727050, 7.40261368023379088714969141554, 8.219999898483099859615488379360, 8.853979073275904708468942658737, 10.13995271856285502447667222050

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