Properties

Label 2-3800-1.1-c1-0-40
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  + 0.848·3-s + 1.74·7-s − 2.28·9-s + 5.92·11-s + 6.78·13-s + 1.86·17-s − 19-s + 1.48·21-s + 5.94·23-s − 4.47·27-s + 3.29·29-s − 5.75·31-s + 5.02·33-s − 4.36·37-s + 5.75·39-s + 7.12·41-s − 6.98·43-s − 4.02·47-s − 3.95·49-s + 1.57·51-s − 9.19·53-s − 0.848·57-s + 2.51·59-s − 2.49·61-s − 3.97·63-s + 6.90·67-s + 5.04·69-s + ⋯
L(s)  = 1  + 0.489·3-s + 0.659·7-s − 0.760·9-s + 1.78·11-s + 1.88·13-s + 0.451·17-s − 0.229·19-s + 0.322·21-s + 1.23·23-s − 0.862·27-s + 0.612·29-s − 1.03·31-s + 0.874·33-s − 0.717·37-s + 0.921·39-s + 1.11·41-s − 1.06·43-s − 0.587·47-s − 0.565·49-s + 0.220·51-s − 1.26·53-s − 0.112·57-s + 0.327·59-s − 0.319·61-s − 0.501·63-s + 0.843·67-s + 0.606·69-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

\(L(1)\) \(\approx\) \(2.966089695\)
\(L(\frac12)\) \(\approx\) \(2.966089695\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.848T + 3T^{2} \)
7 \( 1 - 1.74T + 7T^{2} \)
11 \( 1 - 5.92T + 11T^{2} \)
13 \( 1 - 6.78T + 13T^{2} \)
17 \( 1 - 1.86T + 17T^{2} \)
23 \( 1 - 5.94T + 23T^{2} \)
29 \( 1 - 3.29T + 29T^{2} \)
31 \( 1 + 5.75T + 31T^{2} \)
37 \( 1 + 4.36T + 37T^{2} \)
41 \( 1 - 7.12T + 41T^{2} \)
43 \( 1 + 6.98T + 43T^{2} \)
47 \( 1 + 4.02T + 47T^{2} \)
53 \( 1 + 9.19T + 53T^{2} \)
59 \( 1 - 2.51T + 59T^{2} \)
61 \( 1 + 2.49T + 61T^{2} \)
67 \( 1 - 6.90T + 67T^{2} \)
71 \( 1 - 1.27T + 71T^{2} \)
73 \( 1 - 12.1T + 73T^{2} \)
79 \( 1 + 13.8T + 79T^{2} \)
83 \( 1 + 4.94T + 83T^{2} \)
89 \( 1 - 15.6T + 89T^{2} \)
97 \( 1 + 15.7T + 97T^{2} \)
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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

−8.556083496164942584725183680011, −8.030850053255728568508737705346, −6.94084540561536914241874896911, −6.32194987077122756093635761032, −5.60178943150960447263163578974, −4.61701829129899550253402431991, −3.64822751435999229273070736384, −3.24856094615183467759307554723, −1.81764494964424915577628759079, −1.09070139763871223489935896851, 1.09070139763871223489935896851, 1.81764494964424915577628759079, 3.24856094615183467759307554723, 3.64822751435999229273070736384, 4.61701829129899550253402431991, 5.60178943150960447263163578974, 6.32194987077122756093635761032, 6.94084540561536914241874896911, 8.030850053255728568508737705346, 8.556083496164942584725183680011

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