Properties

Label 2-1900-95.94-c2-0-2
Degree $2$
Conductor $1900$
Sign $-0.447 + 0.894i$
Analytic cond. $51.7712$
Root an. cond. $7.19522$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 13.8i·7-s − 9·9-s − 20.3·11-s + 18.9i·17-s + 19·19-s + 30i·23-s − 53.8i·43-s − 86.5i·47-s − 142.·49-s + 5.12·61-s − 124. i·63-s + 112. i·73-s − 281. i·77-s + 81·81-s − 90i·83-s + ⋯
L(s)  = 1  + 1.97i·7-s − 9-s − 1.85·11-s + 1.11i·17-s + 19-s + 1.30i·23-s − 1.25i·43-s − 1.84i·47-s − 2.90·49-s + 0.0839·61-s − 1.97i·63-s + 1.53i·73-s − 3.65i·77-s + 81-s − 1.08i·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1900\)    =    \(2^{2} \cdot 5^{2} \cdot 19\)
Sign: $-0.447 + 0.894i$
Analytic conductor: \(51.7712\)
Root analytic conductor: \(7.19522\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1900} (949, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1900,\ (\ :1),\ -0.447 + 0.894i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.1711048888\)
\(L(\frac12)\) \(\approx\) \(0.1711048888\)
\(L(2)\) 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 \( 1 - 19T \)
good3 \( 1 + 9T^{2} \)
7 \( 1 - 13.8iT - 49T^{2} \)
11 \( 1 + 20.3T + 121T^{2} \)
13 \( 1 + 169T^{2} \)
17 \( 1 - 18.9iT - 289T^{2} \)
23 \( 1 - 30iT - 529T^{2} \)
29 \( 1 - 841T^{2} \)
31 \( 1 - 961T^{2} \)
37 \( 1 + 1.36e3T^{2} \)
41 \( 1 - 1.68e3T^{2} \)
43 \( 1 + 53.8iT - 1.84e3T^{2} \)
47 \( 1 + 86.5iT - 2.20e3T^{2} \)
53 \( 1 + 2.80e3T^{2} \)
59 \( 1 - 3.48e3T^{2} \)
61 \( 1 - 5.12T + 3.72e3T^{2} \)
67 \( 1 + 4.48e3T^{2} \)
71 \( 1 - 5.04e3T^{2} \)
73 \( 1 - 112. iT - 5.32e3T^{2} \)
79 \( 1 - 6.24e3T^{2} \)
83 \( 1 + 90iT - 6.88e3T^{2} \)
89 \( 1 - 7.92e3T^{2} \)
97 \( 1 + 9.40e3T^{2} \)
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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

−9.470014664021291536672263109998, −8.574103634856952399680217366196, −8.275415144612624581953258678977, −7.35492251613932384590414334980, −6.05458021262539009195781219428, −5.41008814934566540657043998432, −5.23750641639500896400928170143, −3.46955091150134733802647191802, −2.69919398369306086887951836987, −1.95772832957973381091299841992, 0.05172592597765872482746506950, 0.903643945131157991646526320004, 2.63631588612889176957320391162, 3.26358914022313266706201003732, 4.53924545220920111612017835168, 5.06669952269816584851684752958, 6.13175425257910041081853893149, 7.12246554768381638929393713368, 7.72918898041126873644995136718, 8.222867898449766359484820575921

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