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Properties

Label 2-950-25.21-c1-0-27
Degree $2$
Conductor $950$
Sign $0.825 + 0.564i$
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  + (0.309 − 0.951i)2-s + (0.380 + 0.276i)3-s + (−0.809 − 0.587i)4-s + (1.10 + 1.94i)5-s + (0.380 − 0.276i)6-s + 4.66·7-s + (−0.809 + 0.587i)8-s + (−0.858 − 2.64i)9-s + (2.19 − 0.450i)10-s + (1.40 − 4.33i)11-s + (−0.145 − 0.446i)12-s + (1.30 + 4.02i)13-s + (1.44 − 4.43i)14-s + (−0.116 + 1.04i)15-s + (0.309 + 0.951i)16-s + (−2.71 + 1.97i)17-s + ⋯
L(s)  = 1  + (0.218 − 0.672i)2-s + (0.219 + 0.159i)3-s + (−0.404 − 0.293i)4-s + (0.494 + 0.869i)5-s + (0.155 − 0.112i)6-s + 1.76·7-s + (−0.286 + 0.207i)8-s + (−0.286 − 0.881i)9-s + (0.692 − 0.142i)10-s + (0.424 − 1.30i)11-s + (−0.0419 − 0.129i)12-s + (0.362 + 1.11i)13-s + (0.385 − 1.18i)14-s + (−0.0301 + 0.269i)15-s + (0.0772 + 0.237i)16-s + (−0.659 + 0.479i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.825 + 0.564i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.825 + 0.564i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $0.825 + 0.564i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (571, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ 0.825 + 0.564i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.28485 - 0.706767i\)
\(L(\frac12)\) \(\approx\) \(2.28485 - 0.706767i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.309 + 0.951i)T \)
5 \( 1 + (-1.10 - 1.94i)T \)
19 \( 1 + (-0.809 + 0.587i)T \)
good3 \( 1 + (-0.380 - 0.276i)T + (0.927 + 2.85i)T^{2} \)
7 \( 1 - 4.66T + 7T^{2} \)
11 \( 1 + (-1.40 + 4.33i)T + (-8.89 - 6.46i)T^{2} \)
13 \( 1 + (-1.30 - 4.02i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (2.71 - 1.97i)T + (5.25 - 16.1i)T^{2} \)
23 \( 1 + (1.85 - 5.70i)T + (-18.6 - 13.5i)T^{2} \)
29 \( 1 + (2.29 + 1.66i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-7.84 + 5.70i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (0.919 + 2.82i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (0.861 + 2.65i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 3.28T + 43T^{2} \)
47 \( 1 + (-3.35 - 2.43i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (4.11 + 2.98i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-2.33 - 7.18i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (0.689 - 2.12i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (-3.87 + 2.81i)T + (20.7 - 63.7i)T^{2} \)
71 \( 1 + (5.38 + 3.91i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (3.68 - 11.3i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (9.99 + 7.26i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (1.50 - 1.09i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + (-5.42 + 16.7i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (-6.95 - 5.04i)T + (29.9 + 92.2i)T^{2} \)
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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

−10.06723170983668550291368146511, −9.086357175222882594832259682898, −8.599611537685334806632689814792, −7.52006079285556842741987266739, −6.26643572858447282506500066653, −5.70599317516397317508510496754, −4.33517453253963023785214767487, −3.63594674878052113234817283568, −2.39813751197053935408122431619, −1.35482111288575577117526876829, 1.39231113291065151148369573862, 2.46008868901266727610987594302, 4.43241099865027512521840046825, 4.84811008314036732075369099666, 5.55109731482337609947459705496, 6.79363915467993933193683454718, 7.84637876189938179546043244923, 8.260752163778905914876823552744, 8.930549186316957351815822234085, 10.07226264602191717397922942170

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