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Properties

Label 2-950-19.16-c1-0-19
Degree $2$
Conductor $950$
Sign $-0.479 + 0.877i$
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  + (−0.766 − 0.642i)2-s + (−0.476 + 0.173i)3-s + (0.173 + 0.984i)4-s + (0.476 + 0.173i)6-s + (0.753 + 1.30i)7-s + (0.500 − 0.866i)8-s + (−2.10 + 1.76i)9-s + (1.03 − 1.79i)11-s + (−0.253 − 0.439i)12-s + (−5.26 − 1.91i)13-s + (0.261 − 1.48i)14-s + (−0.939 + 0.342i)16-s + (−2.67 − 2.24i)17-s + 2.74·18-s + (3.36 + 2.77i)19-s + ⋯
L(s)  = 1  + (−0.541 − 0.454i)2-s + (−0.275 + 0.100i)3-s + (0.0868 + 0.492i)4-s + (0.194 + 0.0708i)6-s + (0.284 + 0.493i)7-s + (0.176 − 0.306i)8-s + (−0.700 + 0.587i)9-s + (0.312 − 0.540i)11-s + (−0.0732 − 0.126i)12-s + (−1.45 − 0.531i)13-s + (0.0699 − 0.396i)14-s + (−0.234 + 0.0855i)16-s + (−0.648 − 0.544i)17-s + 0.646·18-s + (0.771 + 0.636i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.479 + 0.877i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.479 + 0.877i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $-0.479 + 0.877i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (301, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ -0.479 + 0.877i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.303114 - 0.510881i\)
\(L(\frac12)\) \(\approx\) \(0.303114 - 0.510881i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.766 + 0.642i)T \)
5 \( 1 \)
19 \( 1 + (-3.36 - 2.77i)T \)
good3 \( 1 + (0.476 - 0.173i)T + (2.29 - 1.92i)T^{2} \)
7 \( 1 + (-0.753 - 1.30i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.03 + 1.79i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (5.26 + 1.91i)T + (9.95 + 8.35i)T^{2} \)
17 \( 1 + (2.67 + 2.24i)T + (2.95 + 16.7i)T^{2} \)
23 \( 1 + (1.20 + 6.83i)T + (-21.6 + 7.86i)T^{2} \)
29 \( 1 + (-6.38 + 5.35i)T + (5.03 - 28.5i)T^{2} \)
31 \( 1 + (-0.844 - 1.46i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 0.297T + 37T^{2} \)
41 \( 1 + (2.94 - 1.07i)T + (31.4 - 26.3i)T^{2} \)
43 \( 1 + (-1.40 + 7.95i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (-3.39 + 2.84i)T + (8.16 - 46.2i)T^{2} \)
53 \( 1 + (-1.25 - 7.10i)T + (-49.8 + 18.1i)T^{2} \)
59 \( 1 + (8.89 + 7.46i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (2.23 + 12.6i)T + (-57.3 + 20.8i)T^{2} \)
67 \( 1 + (7.34 - 6.16i)T + (11.6 - 65.9i)T^{2} \)
71 \( 1 + (-0.508 + 2.88i)T + (-66.7 - 24.2i)T^{2} \)
73 \( 1 + (-5.58 + 2.03i)T + (55.9 - 46.9i)T^{2} \)
79 \( 1 + (11.2 - 4.08i)T + (60.5 - 50.7i)T^{2} \)
83 \( 1 + (1.94 + 3.37i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (5.44 + 1.98i)T + (68.1 + 57.2i)T^{2} \)
97 \( 1 + (12.3 + 10.3i)T + (16.8 + 95.5i)T^{2} \)
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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

−9.932360287857174599120319044898, −8.878601740253581710690307235264, −8.274048026348812659728753850840, −7.45393128450026709041605727662, −6.32983605393929284442277006288, −5.31965302691621882300281634175, −4.51086449871327109574527860025, −2.97010960869478114134206355467, −2.23779390876423324948021544859, −0.35610788078987760720981246297, 1.33654275385900114967288119090, 2.81005663343785369017360490616, 4.31502301317510340956388845886, 5.16238242920406139998401104967, 6.21125783133435134673700103997, 7.08276385238266121655750303410, 7.58277310721576133802657936022, 8.763974123308111732611773054469, 9.406848824484770225230265064017, 10.12712601289688822412925082627

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