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Properties

Label 2-1900-475.56-c0-0-0
Degree $2$
Conductor $1900$
Sign $0.985 - 0.166i$
Analytic cond. $0.948223$
Root an. cond. $0.973767$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.978 + 0.207i)5-s + 1.33·7-s + (0.309 − 0.951i)9-s + (0.413 + 1.27i)11-s + (0.169 + 0.122i)17-s + (−0.809 − 0.587i)19-s + (0.190 + 0.587i)23-s + (0.913 − 0.406i)25-s + (−1.30 + 0.278i)35-s + 1.82·43-s + (−0.104 + 0.994i)45-s + (0.809 − 0.587i)47-s + 0.790·49-s + (−0.669 − 1.15i)55-s + (0.564 + 1.73i)61-s + ⋯
L(s)  = 1  + (−0.978 + 0.207i)5-s + 1.33·7-s + (0.309 − 0.951i)9-s + (0.413 + 1.27i)11-s + (0.169 + 0.122i)17-s + (−0.809 − 0.587i)19-s + (0.190 + 0.587i)23-s + (0.913 − 0.406i)25-s + (−1.30 + 0.278i)35-s + 1.82·43-s + (−0.104 + 0.994i)45-s + (0.809 − 0.587i)47-s + 0.790·49-s + (−0.669 − 1.15i)55-s + (0.564 + 1.73i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 - 0.166i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 - 0.166i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1900\)    =    \(2^{2} \cdot 5^{2} \cdot 19\)
Sign: $0.985 - 0.166i$
Analytic conductor: \(0.948223\)
Root analytic conductor: \(0.973767\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1900} (1481, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1900,\ (\ :0),\ 0.985 - 0.166i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.176076031\)
\(L(\frac12)\) \(\approx\) \(1.176076031\)
\(L(1)\) 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 \( 1 + (0.978 - 0.207i)T \)
19 \( 1 + (0.809 + 0.587i)T \)
good3 \( 1 + (-0.309 + 0.951i)T^{2} \)
7 \( 1 - 1.33T + T^{2} \)
11 \( 1 + (-0.413 - 1.27i)T + (-0.809 + 0.587i)T^{2} \)
13 \( 1 + (0.809 + 0.587i)T^{2} \)
17 \( 1 + (-0.169 - 0.122i)T + (0.309 + 0.951i)T^{2} \)
23 \( 1 + (-0.190 - 0.587i)T + (-0.809 + 0.587i)T^{2} \)
29 \( 1 + (-0.309 + 0.951i)T^{2} \)
31 \( 1 + (-0.309 - 0.951i)T^{2} \)
37 \( 1 + (0.809 + 0.587i)T^{2} \)
41 \( 1 + (0.809 + 0.587i)T^{2} \)
43 \( 1 - 1.82T + T^{2} \)
47 \( 1 + (-0.809 + 0.587i)T + (0.309 - 0.951i)T^{2} \)
53 \( 1 + (-0.309 + 0.951i)T^{2} \)
59 \( 1 + (0.809 + 0.587i)T^{2} \)
61 \( 1 + (-0.564 - 1.73i)T + (-0.809 + 0.587i)T^{2} \)
67 \( 1 + (-0.309 - 0.951i)T^{2} \)
71 \( 1 + (-0.309 + 0.951i)T^{2} \)
73 \( 1 + (0.309 + 0.951i)T + (-0.809 + 0.587i)T^{2} \)
79 \( 1 + (-0.309 + 0.951i)T^{2} \)
83 \( 1 + (-1.30 - 0.951i)T + (0.309 + 0.951i)T^{2} \)
89 \( 1 + (0.809 - 0.587i)T^{2} \)
97 \( 1 + (-0.309 + 0.951i)T^{2} \)
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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

−9.254242572847066652941301822164, −8.677051980695745482023479783054, −7.69380726496772858068207744620, −7.25073382215159963124705388756, −6.44646599450291164715501251745, −5.21185446234379797173808815072, −4.31431059784484800654274182049, −3.91315109061995638198355966103, −2.48194514041375049682327861837, −1.24876908150718269162853781313, 1.14833579578359337394675630989, 2.41685540173557012219518480774, 3.73309554542912994467638384498, 4.46439003170359210412306998205, 5.18953564413803901453692064142, 6.14740014953190471489553379659, 7.29017659197875176957615753451, 7.987089142012235672508628427149, 8.378593952542438719020985962758, 9.106167245165485410945558476077

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