Properties

Label 2-3800-1.1-c1-0-14
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  − 0.414·3-s + 0.414·7-s − 2.82·9-s − 1.41·11-s − 3.82·13-s + 17-s − 19-s − 0.171·21-s − 3.24·23-s + 2.41·27-s − 1.82·29-s + 0.585·31-s + 0.585·33-s + 10.8·37-s + 1.58·39-s + 7.07·41-s − 6.24·43-s + 8·47-s − 6.82·49-s − 0.414·51-s + 3.82·53-s + 0.414·57-s + 11.5·59-s + 0.585·61-s − 1.17·63-s + 8.07·67-s + 1.34·69-s + ⋯
L(s)  = 1  − 0.239·3-s + 0.156·7-s − 0.942·9-s − 0.426·11-s − 1.06·13-s + 0.242·17-s − 0.229·19-s − 0.0374·21-s − 0.676·23-s + 0.464·27-s − 0.339·29-s + 0.105·31-s + 0.101·33-s + 1.78·37-s + 0.253·39-s + 1.10·41-s − 0.951·43-s + 1.16·47-s − 0.975·49-s − 0.0580·51-s + 0.525·53-s + 0.0548·57-s + 1.50·59-s + 0.0750·61-s − 0.147·63-s + 0.986·67-s + 0.161·69-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

\(L(1)\) \(\approx\) \(1.173087898\)
\(L(\frac12)\) \(\approx\) \(1.173087898\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.414T + 3T^{2} \)
7 \( 1 - 0.414T + 7T^{2} \)
11 \( 1 + 1.41T + 11T^{2} \)
13 \( 1 + 3.82T + 13T^{2} \)
17 \( 1 - T + 17T^{2} \)
23 \( 1 + 3.24T + 23T^{2} \)
29 \( 1 + 1.82T + 29T^{2} \)
31 \( 1 - 0.585T + 31T^{2} \)
37 \( 1 - 10.8T + 37T^{2} \)
41 \( 1 - 7.07T + 41T^{2} \)
43 \( 1 + 6.24T + 43T^{2} \)
47 \( 1 - 8T + 47T^{2} \)
53 \( 1 - 3.82T + 53T^{2} \)
59 \( 1 - 11.5T + 59T^{2} \)
61 \( 1 - 0.585T + 61T^{2} \)
67 \( 1 - 8.07T + 67T^{2} \)
71 \( 1 + 11.0T + 71T^{2} \)
73 \( 1 - 8.17T + 73T^{2} \)
79 \( 1 - 4.82T + 79T^{2} \)
83 \( 1 - 14.4T + 83T^{2} \)
89 \( 1 + 12.2T + 89T^{2} \)
97 \( 1 + 3.65T + 97T^{2} \)
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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

−8.332111251559784136442004962044, −7.87002982830762091675834744919, −7.06691848521577099993701016655, −6.15643792310994844843355344565, −5.52775426131471740497381245548, −4.82329190583088995857440252842, −3.93003178651002150024846813935, −2.81811860622178336617157559684, −2.17304263108598225239215893536, −0.60763894153897535356229012661, 0.60763894153897535356229012661, 2.17304263108598225239215893536, 2.81811860622178336617157559684, 3.93003178651002150024846813935, 4.82329190583088995857440252842, 5.52775426131471740497381245548, 6.15643792310994844843355344565, 7.06691848521577099993701016655, 7.87002982830762091675834744919, 8.332111251559784136442004962044

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