Properties

Label 2-3800-1.1-c1-0-2
Degree $2$
Conductor $3800$
Sign $1$
Analytic cond. $30.3431$
Root an. cond. $5.50846$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.77·3-s − 2.31·7-s + 4.67·9-s + 2.16·11-s − 6.25·13-s − 7.10·17-s − 19-s + 6.40·21-s + 8.99·23-s − 4.63·27-s + 3.16·29-s − 9.95·31-s − 5.98·33-s − 9.43·37-s + 17.3·39-s − 10.8·41-s − 1.05·43-s + 6.98·47-s − 1.66·49-s + 19.6·51-s + 2.69·53-s + 2.77·57-s + 11.2·59-s − 4.36·61-s − 10.7·63-s − 6.25·67-s − 24.9·69-s + ⋯
L(s)  = 1  − 1.59·3-s − 0.873·7-s + 1.55·9-s + 0.651·11-s − 1.73·13-s − 1.72·17-s − 0.229·19-s + 1.39·21-s + 1.87·23-s − 0.892·27-s + 0.587·29-s − 1.78·31-s − 1.04·33-s − 1.55·37-s + 2.77·39-s − 1.68·41-s − 0.160·43-s + 1.01·47-s − 0.237·49-s + 2.75·51-s + 0.370·53-s + 0.366·57-s + 1.45·59-s − 0.558·61-s − 1.36·63-s − 0.764·67-s − 3.00·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3800\)    =    \(2^{3} \cdot 5^{2} \cdot 19\)
Sign: $1$
Analytic conductor: \(30.3431\)
Root analytic conductor: \(5.50846\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3800} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3800,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3453852440\)
\(L(\frac12)\) \(\approx\) \(0.3453852440\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
good3 \( 1 + 2.77T + 3T^{2} \)
7 \( 1 + 2.31T + 7T^{2} \)
11 \( 1 - 2.16T + 11T^{2} \)
13 \( 1 + 6.25T + 13T^{2} \)
17 \( 1 + 7.10T + 17T^{2} \)
23 \( 1 - 8.99T + 23T^{2} \)
29 \( 1 - 3.16T + 29T^{2} \)
31 \( 1 + 9.95T + 31T^{2} \)
37 \( 1 + 9.43T + 37T^{2} \)
41 \( 1 + 10.8T + 41T^{2} \)
43 \( 1 + 1.05T + 43T^{2} \)
47 \( 1 - 6.98T + 47T^{2} \)
53 \( 1 - 2.69T + 53T^{2} \)
59 \( 1 - 11.2T + 59T^{2} \)
61 \( 1 + 4.36T + 61T^{2} \)
67 \( 1 + 6.25T + 67T^{2} \)
71 \( 1 + 13.9T + 71T^{2} \)
73 \( 1 - 8.64T + 73T^{2} \)
79 \( 1 + 9.93T + 79T^{2} \)
83 \( 1 - 9.82T + 83T^{2} \)
89 \( 1 + 1.63T + 89T^{2} \)
97 \( 1 + 9.27T + 97T^{2} \)
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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

−8.789820957144421385365500408544, −7.17017583187146722531641021754, −6.96766724839201932336989052013, −6.43133618049960926916116571016, −5.38180299476045708055427866309, −4.94913892662016626665419444919, −4.13553955410216834044883967172, −2.99314187427319959733149592097, −1.81450089913514348343164485599, −0.35630216519218056329676458585, 0.35630216519218056329676458585, 1.81450089913514348343164485599, 2.99314187427319959733149592097, 4.13553955410216834044883967172, 4.94913892662016626665419444919, 5.38180299476045708055427866309, 6.43133618049960926916116571016, 6.96766724839201932336989052013, 7.17017583187146722531641021754, 8.789820957144421385365500408544

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