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Properties

Label 2-950-95.78-c1-0-15
Degree $2$
Conductor $950$
Sign $-0.786 + 0.617i$
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  + (−0.819 + 0.573i)2-s + (−2.17 + 0.190i)3-s + (0.342 − 0.939i)4-s + (1.67 − 1.40i)6-s + (−0.431 − 0.115i)7-s + (0.258 + 0.965i)8-s + (1.74 − 0.307i)9-s + (2.37 − 4.12i)11-s + (−0.565 + 2.11i)12-s + (−0.210 + 2.40i)13-s + (0.419 − 0.152i)14-s + (−0.766 − 0.642i)16-s + (−0.392 − 0.560i)17-s + (−1.25 + 1.25i)18-s + (−2.15 + 3.79i)19-s + ⋯
L(s)  = 1  + (−0.579 + 0.405i)2-s + (−1.25 + 0.109i)3-s + (0.171 − 0.469i)4-s + (0.683 − 0.573i)6-s + (−0.163 − 0.0436i)7-s + (0.0915 + 0.341i)8-s + (0.582 − 0.102i)9-s + (0.717 − 1.24i)11-s + (−0.163 + 0.609i)12-s + (−0.0582 + 0.666i)13-s + (0.112 − 0.0408i)14-s + (−0.191 − 0.160i)16-s + (−0.0951 − 0.135i)17-s + (−0.295 + 0.295i)18-s + (−0.493 + 0.869i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.786 + 0.617i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.786 + 0.617i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $-0.786 + 0.617i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (743, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ -0.786 + 0.617i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0282300 - 0.0816493i\)
\(L(\frac12)\) \(\approx\) \(0.0282300 - 0.0816493i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.819 - 0.573i)T \)
5 \( 1 \)
19 \( 1 + (2.15 - 3.79i)T \)
good3 \( 1 + (2.17 - 0.190i)T + (2.95 - 0.520i)T^{2} \)
7 \( 1 + (0.431 + 0.115i)T + (6.06 + 3.5i)T^{2} \)
11 \( 1 + (-2.37 + 4.12i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.210 - 2.40i)T + (-12.8 - 2.25i)T^{2} \)
17 \( 1 + (0.392 + 0.560i)T + (-5.81 + 15.9i)T^{2} \)
23 \( 1 + (-0.188 - 0.404i)T + (-14.7 + 17.6i)T^{2} \)
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 \( 1 + (8.46 + 3.94i)T + (27.6 + 32.9i)T^{2} \)
47 \( 1 + (4.32 + 3.02i)T + (16.0 + 44.1i)T^{2} \)
53 \( 1 + (7.35 - 3.42i)T + (34.0 - 40.6i)T^{2} \)
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 \( 1 + (-6.44 + 9.20i)T + (-22.9 - 62.9i)T^{2} \)
71 \( 1 + (-0.879 - 2.41i)T + (-54.3 + 45.6i)T^{2} \)
73 \( 1 + (0.378 + 4.32i)T + (-71.8 + 12.6i)T^{2} \)
79 \( 1 + (12.3 + 10.3i)T + (13.7 + 77.7i)T^{2} \)
83 \( 1 + (0.224 - 0.838i)T + (-71.8 - 41.5i)T^{2} \)
89 \( 1 + (12.7 - 10.7i)T + (15.4 - 87.6i)T^{2} \)
97 \( 1 + (2.46 - 1.72i)T + (33.1 - 91.1i)T^{2} \)
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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

−9.769774241506197588911000213399, −8.841643098095082466072365999031, −8.146637765209157691888826387855, −6.83742385406012273988700805604, −6.35458205592210015537306567905, −5.61237845700130518835593565917, −4.65985169730610410121003065648, −3.38878566381122602610228442609, −1.50814605729613405698772743271, −0.06230709753686718775002027186, 1.41067632958633142131461986039, 2.82476556017758423483246442334, 4.29885968443936372636735492778, 5.09450192563897594959763001639, 6.33801831776526319103570785515, 6.79365853281959948145192722647, 7.85183889137858708842267298605, 8.831390520108284405964941757038, 9.832712784814461413658845167460, 10.28414339544930506994027072612

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