Properties

Label 2-3800-1.1-c1-0-19
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  − 1.93·3-s + 1.24·7-s + 0.747·9-s − 0.513·11-s + 6.15·13-s − 4.51·17-s − 19-s − 2.41·21-s − 5.86·23-s + 4.36·27-s + 6.62·29-s + 6.41·31-s + 0.994·33-s − 1.40·37-s − 11.9·39-s + 10.6·41-s − 3.04·43-s − 1.99·47-s − 5.44·49-s + 8.74·51-s − 14.0·53-s + 1.93·57-s − 4.34·59-s + 10.7·61-s + 0.932·63-s − 9.89·67-s + 11.3·69-s + ⋯
L(s)  = 1  − 1.11·3-s + 0.471·7-s + 0.249·9-s − 0.154·11-s + 1.70·13-s − 1.09·17-s − 0.229·19-s − 0.526·21-s − 1.22·23-s + 0.839·27-s + 1.23·29-s + 1.15·31-s + 0.173·33-s − 0.231·37-s − 1.90·39-s + 1.66·41-s − 0.464·43-s − 0.290·47-s − 0.777·49-s + 1.22·51-s − 1.93·53-s + 0.256·57-s − 0.565·59-s + 1.37·61-s + 0.117·63-s − 1.20·67-s + 1.36·69-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

\(L(1)\) \(\approx\) \(1.187856709\)
\(L(\frac12)\) \(\approx\) \(1.187856709\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.93T + 3T^{2} \)
7 \( 1 - 1.24T + 7T^{2} \)
11 \( 1 + 0.513T + 11T^{2} \)
13 \( 1 - 6.15T + 13T^{2} \)
17 \( 1 + 4.51T + 17T^{2} \)
23 \( 1 + 5.86T + 23T^{2} \)
29 \( 1 - 6.62T + 29T^{2} \)
31 \( 1 - 6.41T + 31T^{2} \)
37 \( 1 + 1.40T + 37T^{2} \)
41 \( 1 - 10.6T + 41T^{2} \)
43 \( 1 + 3.04T + 43T^{2} \)
47 \( 1 + 1.99T + 47T^{2} \)
53 \( 1 + 14.0T + 53T^{2} \)
59 \( 1 + 4.34T + 59T^{2} \)
61 \( 1 - 10.7T + 61T^{2} \)
67 \( 1 + 9.89T + 67T^{2} \)
71 \( 1 - 7.42T + 71T^{2} \)
73 \( 1 - 12.8T + 73T^{2} \)
79 \( 1 - 2.56T + 79T^{2} \)
83 \( 1 + 7.50T + 83T^{2} \)
89 \( 1 + 7.85T + 89T^{2} \)
97 \( 1 - 6.74T + 97T^{2} \)
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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

−8.329969858292361094229887419795, −7.983451765466953643050475616939, −6.50284192652144108068409233795, −6.44479818527058019354710437964, −5.61983152234551653815331344414, −4.71292219796688939119103390979, −4.15706055029957661762160157046, −2.98833209566256860990705508822, −1.77939889325735359663450105121, −0.68198939034702020857083765837, 0.68198939034702020857083765837, 1.77939889325735359663450105121, 2.98833209566256860990705508822, 4.15706055029957661762160157046, 4.71292219796688939119103390979, 5.61983152234551653815331344414, 6.44479818527058019354710437964, 6.50284192652144108068409233795, 7.983451765466953643050475616939, 8.329969858292361094229887419795

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