Properties

Label 2-3800-1.1-c1-0-11
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  + 1.08·3-s − 4.19·7-s − 1.81·9-s − 6.43·11-s − 2.24·13-s + 7.84·17-s − 19-s − 4.57·21-s + 0.859·23-s − 5.24·27-s + 8.38·29-s + 1.24·31-s − 7.00·33-s + 6.79·37-s − 2.44·39-s + 5.92·41-s − 6.81·43-s + 6.00·47-s + 10.6·49-s + 8.55·51-s − 13.7·53-s − 1.08·57-s + 6.88·59-s − 1.31·61-s + 7.60·63-s − 4.73·67-s + 0.936·69-s + ⋯
L(s)  = 1  + 0.629·3-s − 1.58·7-s − 0.603·9-s − 1.93·11-s − 0.622·13-s + 1.90·17-s − 0.229·19-s − 0.998·21-s + 0.179·23-s − 1.00·27-s + 1.55·29-s + 0.223·31-s − 1.22·33-s + 1.11·37-s − 0.392·39-s + 0.925·41-s − 1.03·43-s + 0.876·47-s + 1.51·49-s + 1.19·51-s − 1.88·53-s − 0.144·57-s + 0.896·59-s − 0.168·61-s + 0.958·63-s − 0.578·67-s + 0.112·69-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

\(L(1)\) \(\approx\) \(1.230594208\)
\(L(\frac12)\) \(\approx\) \(1.230594208\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.08T + 3T^{2} \)
7 \( 1 + 4.19T + 7T^{2} \)
11 \( 1 + 6.43T + 11T^{2} \)
13 \( 1 + 2.24T + 13T^{2} \)
17 \( 1 - 7.84T + 17T^{2} \)
23 \( 1 - 0.859T + 23T^{2} \)
29 \( 1 - 8.38T + 29T^{2} \)
31 \( 1 - 1.24T + 31T^{2} \)
37 \( 1 - 6.79T + 37T^{2} \)
41 \( 1 - 5.92T + 41T^{2} \)
43 \( 1 + 6.81T + 43T^{2} \)
47 \( 1 - 6.00T + 47T^{2} \)
53 \( 1 + 13.7T + 53T^{2} \)
59 \( 1 - 6.88T + 59T^{2} \)
61 \( 1 + 1.31T + 61T^{2} \)
67 \( 1 + 4.73T + 67T^{2} \)
71 \( 1 - 10.2T + 71T^{2} \)
73 \( 1 - 9.86T + 73T^{2} \)
79 \( 1 - 6.47T + 79T^{2} \)
83 \( 1 + 5.42T + 83T^{2} \)
89 \( 1 - 3.10T + 89T^{2} \)
97 \( 1 + 6.76T + 97T^{2} \)
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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

−8.260527061686267909731442160855, −7.950136948845861959628957833212, −7.16556119509388174887259787848, −6.19313416139401745224242988340, −5.58741050448752143219939993653, −4.79735791811941317414249959678, −3.49013750588492898404309316889, −2.91657404336742598428573155388, −2.47561651512472468204297282998, −0.58249917694669068976413874887, 0.58249917694669068976413874887, 2.47561651512472468204297282998, 2.91657404336742598428573155388, 3.49013750588492898404309316889, 4.79735791811941317414249959678, 5.58741050448752143219939993653, 6.19313416139401745224242988340, 7.16556119509388174887259787848, 7.950136948845861959628957833212, 8.260527061686267909731442160855

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