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Properties

Label 2-950-19.16-c1-0-3
Degree $2$
Conductor $950$
Sign $-0.813 - 0.581i$
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 + (0.0801 − 0.0291i)3-s + (0.173 + 0.984i)4-s + (0.0801 + 0.0291i)6-s + (−0.920 − 1.59i)7-s + (−0.500 + 0.866i)8-s + (−2.29 + 1.92i)9-s + (−1.21 + 2.09i)11-s + (0.0426 + 0.0738i)12-s + (−2.84 − 1.03i)13-s + (0.319 − 1.81i)14-s + (−0.939 + 0.342i)16-s + (5.22 + 4.38i)17-s − 2.99·18-s + (−3.15 + 3.00i)19-s + ⋯
L(s)  = 1  + (0.541 + 0.454i)2-s + (0.0462 − 0.0168i)3-s + (0.0868 + 0.492i)4-s + (0.0327 + 0.0119i)6-s + (−0.347 − 0.602i)7-s + (−0.176 + 0.306i)8-s + (−0.764 + 0.641i)9-s + (−0.364 + 0.632i)11-s + (0.0123 + 0.0213i)12-s + (−0.790 − 0.287i)13-s + (0.0854 − 0.484i)14-s + (−0.234 + 0.0855i)16-s + (1.26 + 1.06i)17-s − 0.705·18-s + (−0.724 + 0.689i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.813 - 0.581i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.813 - 0.581i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $-0.813 - 0.581i$
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: \((2,\ 950,\ (\ :1/2),\ -0.813 - 0.581i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.385301 + 1.20102i\)
\(L(\frac12)\) \(\approx\) \(0.385301 + 1.20102i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.15 - 3.00i)T \)
good3 \( 1 + (-0.0801 + 0.0291i)T + (2.29 - 1.92i)T^{2} \)
7 \( 1 + (0.920 + 1.59i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.21 - 2.09i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.84 + 1.03i)T + (9.95 + 8.35i)T^{2} \)
17 \( 1 + (-5.22 - 4.38i)T + (2.95 + 16.7i)T^{2} \)
23 \( 1 + (-1.39 - 7.90i)T + (-21.6 + 7.86i)T^{2} \)
29 \( 1 + (7.77 - 6.52i)T + (5.03 - 28.5i)T^{2} \)
31 \( 1 + (2.41 + 4.18i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 0.202T + 37T^{2} \)
41 \( 1 + (-5.41 + 1.97i)T + (31.4 - 26.3i)T^{2} \)
43 \( 1 + (-0.365 + 2.07i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (-7.53 + 6.32i)T + (8.16 - 46.2i)T^{2} \)
53 \( 1 + (-1.08 - 6.17i)T + (-49.8 + 18.1i)T^{2} \)
59 \( 1 + (4.62 + 3.87i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (-0.741 - 4.20i)T + (-57.3 + 20.8i)T^{2} \)
67 \( 1 + (-2.39 + 2.00i)T + (11.6 - 65.9i)T^{2} \)
71 \( 1 + (1.94 - 11.0i)T + (-66.7 - 24.2i)T^{2} \)
73 \( 1 + (-7.33 + 2.67i)T + (55.9 - 46.9i)T^{2} \)
79 \( 1 + (9.87 - 3.59i)T + (60.5 - 50.7i)T^{2} \)
83 \( 1 + (6.37 + 11.0i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (1.21 + 0.442i)T + (68.1 + 57.2i)T^{2} \)
97 \( 1 + (6.28 + 5.27i)T + (16.8 + 95.5i)T^{2} \)
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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

−10.40063810479278034063297347044, −9.601980789681050309753593971364, −8.496819414998342845205800017261, −7.51529225435416938354769638090, −7.29269525271971314480888139380, −5.71156254625411242197676988047, −5.47120212436542712494548236079, −4.11237904121111709400528761072, −3.28889267036857752888572610968, −1.95647865080685587185735763650, 0.45740051293582233720356722030, 2.50126044686278924045995670533, 3.02839975044878009610470980625, 4.30350127679087236278527849895, 5.38229413426233088462716799269, 6.02082239394867636075864999896, 6.99453131584367961370593402667, 8.122003161918596844995652251296, 9.153485887004729230293097984301, 9.575480794281256020705921281159

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