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Properties

Label 2-950-19.9-c1-0-5
Degree $2$
Conductor $950$
Sign $-0.690 + 0.723i$
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  + (0.173 + 0.984i)2-s + (−1.71 + 1.43i)3-s + (−0.939 + 0.342i)4-s + (−1.71 − 1.43i)6-s + (−1.40 + 2.43i)7-s + (−0.5 − 0.866i)8-s + (0.345 − 1.96i)9-s + (3.20 + 5.54i)11-s + (1.11 − 1.93i)12-s + (3.65 + 3.07i)13-s + (−2.64 − 0.963i)14-s + (0.766 − 0.642i)16-s + (1.20 + 6.82i)17-s + 1.99·18-s + (−2.52 − 3.55i)19-s + ⋯
L(s)  = 1  + (0.122 + 0.696i)2-s + (−0.988 + 0.829i)3-s + (−0.469 + 0.171i)4-s + (−0.698 − 0.586i)6-s + (−0.532 + 0.921i)7-s + (−0.176 − 0.306i)8-s + (0.115 − 0.653i)9-s + (0.965 + 1.67i)11-s + (0.322 − 0.558i)12-s + (1.01 + 0.851i)13-s + (−0.707 − 0.257i)14-s + (0.191 − 0.160i)16-s + (0.291 + 1.65i)17-s + 0.469·18-s + (−0.579 − 0.814i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.690 + 0.723i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.690 + 0.723i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $-0.690 + 0.723i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (351, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ -0.690 + 0.723i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.348997 - 0.815705i\)
\(L(\frac12)\) \(\approx\) \(0.348997 - 0.815705i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.173 - 0.984i)T \)
5 \( 1 \)
19 \( 1 + (2.52 + 3.55i)T \)
good3 \( 1 + (1.71 - 1.43i)T + (0.520 - 2.95i)T^{2} \)
7 \( 1 + (1.40 - 2.43i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-3.20 - 5.54i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-3.65 - 3.07i)T + (2.25 + 12.8i)T^{2} \)
17 \( 1 + (-1.20 - 6.82i)T + (-15.9 + 5.81i)T^{2} \)
23 \( 1 + (5.11 - 1.86i)T + (17.6 - 14.7i)T^{2} \)
29 \( 1 + (-1.30 + 7.41i)T + (-27.2 - 9.91i)T^{2} \)
31 \( 1 + (-0.656 + 1.13i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 6.63T + 37T^{2} \)
41 \( 1 + (1.76 - 1.48i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 + (7.80 + 2.83i)T + (32.9 + 27.6i)T^{2} \)
47 \( 1 + (-1.33 + 7.59i)T + (-44.1 - 16.0i)T^{2} \)
53 \( 1 + (-3.28 + 1.19i)T + (40.6 - 34.0i)T^{2} \)
59 \( 1 + (-1.48 - 8.39i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (-2.80 + 1.02i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (1.56 - 8.84i)T + (-62.9 - 22.9i)T^{2} \)
71 \( 1 + (-2.34 - 0.851i)T + (54.3 + 45.6i)T^{2} \)
73 \( 1 + (-5.37 + 4.50i)T + (12.6 - 71.8i)T^{2} \)
79 \( 1 + (2.36 - 1.98i)T + (13.7 - 77.7i)T^{2} \)
83 \( 1 + (0.185 - 0.320i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (0.130 + 0.109i)T + (15.4 + 87.6i)T^{2} \)
97 \( 1 + (2.55 + 14.4i)T + (-91.1 + 33.1i)T^{2} \)
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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

−10.30962518396935913907848282917, −9.767992627237400339957610435293, −8.993943027449070188974149488587, −8.128543503891363075506301203784, −6.74733236070673524173446749846, −6.23095802171186247950294227886, −5.55649501094320208436024132524, −4.27225902981256002970779699868, −4.04824441276856808552584103262, −1.99268490698203147991663493729, 0.52666561409918638466546250479, 1.22646636339222642010448938369, 3.15806665556412054466059840190, 3.84426308392328828444786944925, 5.27777949620198912149845495673, 6.17417065097394390599432133446, 6.62468662207911768658697661417, 7.82869558002139760815607020468, 8.704170779414793780038043925935, 9.740408293637259423275094506723

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