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Properties

Label 2-950-25.6-c1-0-28
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: $\chi_{950} (381, \cdot )$
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

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

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