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Properties

Label 2-1900-19.7-c1-0-17
Degree $2$
Conductor $1900$
Sign $0.440 + 0.897i$
Analytic cond. $15.1715$
Root an. cond. $3.89507$
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  + (0.226 + 0.392i)3-s − 2.54·7-s + (1.39 − 2.42i)9-s + 2.22·11-s + (−3.51 + 6.08i)13-s + (−1.27 − 2.21i)17-s + (−2.70 − 3.41i)19-s + (−0.575 − 0.997i)21-s + (4.01 − 6.95i)23-s + 2.62·27-s + (0.941 − 1.63i)29-s + 5.98·31-s + (0.503 + 0.872i)33-s + 2.86·37-s − 3.17·39-s + ⋯
L(s)  = 1  + (0.130 + 0.226i)3-s − 0.961·7-s + (0.465 − 0.806i)9-s + 0.670·11-s + (−0.973 + 1.68i)13-s + (−0.310 − 0.537i)17-s + (−0.620 − 0.784i)19-s + (−0.125 − 0.217i)21-s + (0.837 − 1.44i)23-s + 0.505·27-s + (0.174 − 0.302i)29-s + 1.07·31-s + (0.0876 + 0.151i)33-s + 0.470·37-s − 0.509·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.440 + 0.897i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.440 + 0.897i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1900\)    =    \(2^{2} \cdot 5^{2} \cdot 19\)
Sign: $0.440 + 0.897i$
Analytic conductor: \(15.1715\)
Root analytic conductor: \(3.89507\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1900} (501, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1900,\ (\ :1/2),\ 0.440 + 0.897i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.341452329\)
\(L(\frac12)\) \(\approx\) \(1.341452329\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
19 \( 1 + (2.70 + 3.41i)T \)
good3 \( 1 + (-0.226 - 0.392i)T + (-1.5 + 2.59i)T^{2} \)
7 \( 1 + 2.54T + 7T^{2} \)
11 \( 1 - 2.22T + 11T^{2} \)
13 \( 1 + (3.51 - 6.08i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (1.27 + 2.21i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (-4.01 + 6.95i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.941 + 1.63i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 5.98T + 31T^{2} \)
37 \( 1 - 2.86T + 37T^{2} \)
41 \( 1 + (3.67 + 6.36i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.84 - 3.19i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-2.36 + 4.09i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-5.14 + 8.91i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.73 - 6.47i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.17 + 7.23i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.17 - 10.7i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (4.13 + 7.16i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (6.32 + 10.9i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-2.13 - 3.69i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 14.7T + 83T^{2} \)
89 \( 1 + (-7.19 + 12.4i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (2.91 + 5.04i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.024215901478451985386075191109, −8.727824773009197420030170938640, −7.02811436034026860882736485723, −6.87781991340906955801142121921, −6.17343900733171035116394961141, −4.62035439326148836819363415498, −4.30782678980436673383415817842, −3.13059789724252463109510388978, −2.18033555829990323147410864727, −0.52500707971126980228598242998, 1.20509539312166740212599407145, 2.54581326813441706751142848398, 3.36005873496883264741079025758, 4.42379557412853289139977334673, 5.40234644690001867760861247455, 6.19095046644701767555144991619, 7.08729570098740716550863125815, 7.74458137180699814969003744564, 8.469907310309028585893291296110, 9.465524792590529586910965368209

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