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Properties

Label 2-950-19.16-c1-0-15
Degree $2$
Conductor $950$
Sign $0.299 - 0.954i$
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 + (1.68 − 0.613i)3-s + (0.173 + 0.984i)4-s + (1.68 + 0.613i)6-s + (0.680 + 1.17i)7-s + (−0.500 + 0.866i)8-s + (0.169 − 0.142i)9-s + (−3.22 + 5.59i)11-s + (0.897 + 1.55i)12-s + (5.52 + 2.01i)13-s + (−0.236 + 1.34i)14-s + (−0.939 + 0.342i)16-s + (1.96 + 1.64i)17-s + 0.221·18-s + (−3.83 − 2.06i)19-s + ⋯
L(s)  = 1  + (0.541 + 0.454i)2-s + (0.973 − 0.354i)3-s + (0.0868 + 0.492i)4-s + (0.688 + 0.250i)6-s + (0.257 + 0.445i)7-s + (−0.176 + 0.306i)8-s + (0.0566 − 0.0474i)9-s + (−0.973 + 1.68i)11-s + (0.259 + 0.448i)12-s + (1.53 + 0.557i)13-s + (−0.0631 + 0.358i)14-s + (−0.234 + 0.0855i)16-s + (0.475 + 0.398i)17-s + 0.0522·18-s + (−0.880 − 0.473i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.299 - 0.954i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.299 - 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $0.299 - 0.954i$
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: \((2,\ 950,\ (\ :1/2),\ 0.299 - 0.954i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.30529 + 1.69294i\)
\(L(\frac12)\) \(\approx\) \(2.30529 + 1.69294i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.83 + 2.06i)T \)
good3 \( 1 + (-1.68 + 0.613i)T + (2.29 - 1.92i)T^{2} \)
7 \( 1 + (-0.680 - 1.17i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (3.22 - 5.59i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-5.52 - 2.01i)T + (9.95 + 8.35i)T^{2} \)
17 \( 1 + (-1.96 - 1.64i)T + (2.95 + 16.7i)T^{2} \)
23 \( 1 + (1.46 + 8.29i)T + (-21.6 + 7.86i)T^{2} \)
29 \( 1 + (-3.41 + 2.86i)T + (5.03 - 28.5i)T^{2} \)
31 \( 1 + (-0.701 - 1.21i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 4.42T + 37T^{2} \)
41 \( 1 + (-5.15 + 1.87i)T + (31.4 - 26.3i)T^{2} \)
43 \( 1 + (1.19 - 6.74i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (-7.29 + 6.11i)T + (8.16 - 46.2i)T^{2} \)
53 \( 1 + (0.768 + 4.35i)T + (-49.8 + 18.1i)T^{2} \)
59 \( 1 + (7.87 + 6.60i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (-1.12 - 6.36i)T + (-57.3 + 20.8i)T^{2} \)
67 \( 1 + (-5.14 + 4.31i)T + (11.6 - 65.9i)T^{2} \)
71 \( 1 + (-0.336 + 1.90i)T + (-66.7 - 24.2i)T^{2} \)
73 \( 1 + (5.90 - 2.14i)T + (55.9 - 46.9i)T^{2} \)
79 \( 1 + (-3.37 + 1.22i)T + (60.5 - 50.7i)T^{2} \)
83 \( 1 + (6.74 + 11.6i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-1.78 - 0.648i)T + (68.1 + 57.2i)T^{2} \)
97 \( 1 + (-0.0945 - 0.0793i)T + (16.8 + 95.5i)T^{2} \)
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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

−10.19237974848685956584628285534, −9.008941674316722820238478285857, −8.374179255238674033930967228517, −7.81899279103036862653195275557, −6.82813430367256774903188681436, −6.01006194972324240874749009776, −4.82025740819291553377361040547, −4.05245725671380863660190548072, −2.67874387925679363589802795599, −2.00042138201496340483497737820, 1.06934220494613834236809765604, 2.75556995236670894410497739129, 3.42000710100278945348092496954, 4.13706861680181946056969178804, 5.63262510565624459259551557340, 6.01648976232041594601157163347, 7.65803266535972881056610121248, 8.292018578549791799468884266101, 8.971533342716661820317021959870, 9.971506475717905888132014386428

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