Properties

Label 2-3800-1.1-c1-0-1
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  − 3.26·3-s − 4.07·7-s + 7.63·9-s − 0.786·11-s − 1.07·13-s + 1.90·17-s − 19-s + 13.2·21-s + 1.41·23-s − 15.1·27-s − 7.26·29-s + 2.22·31-s + 2.56·33-s − 9.14·37-s + 3.52·39-s − 6.11·41-s − 8.40·43-s − 3.56·47-s + 9.58·49-s − 6.22·51-s + 8.57·53-s + 3.26·57-s − 13.4·59-s + 12.7·61-s − 31.0·63-s + 5.10·67-s − 4.61·69-s + ⋯
L(s)  = 1  − 1.88·3-s − 1.53·7-s + 2.54·9-s − 0.237·11-s − 0.299·13-s + 0.462·17-s − 0.229·19-s + 2.89·21-s + 0.294·23-s − 2.91·27-s − 1.34·29-s + 0.398·31-s + 0.446·33-s − 1.50·37-s + 0.563·39-s − 0.955·41-s − 1.28·43-s − 0.519·47-s + 1.36·49-s − 0.871·51-s + 1.17·53-s + 0.431·57-s − 1.74·59-s + 1.63·61-s − 3.91·63-s + 0.623·67-s − 0.555·69-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

\(L(1)\) \(\approx\) \(0.3035832069\)
\(L(\frac12)\) \(\approx\) \(0.3035832069\)
\(L(\frac{3}{2})\) 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 \( 1 + 3.26T + 3T^{2} \)
7 \( 1 + 4.07T + 7T^{2} \)
11 \( 1 + 0.786T + 11T^{2} \)
13 \( 1 + 1.07T + 13T^{2} \)
17 \( 1 - 1.90T + 17T^{2} \)
23 \( 1 - 1.41T + 23T^{2} \)
29 \( 1 + 7.26T + 29T^{2} \)
31 \( 1 - 2.22T + 31T^{2} \)
37 \( 1 + 9.14T + 37T^{2} \)
41 \( 1 + 6.11T + 41T^{2} \)
43 \( 1 + 8.40T + 43T^{2} \)
47 \( 1 + 3.56T + 47T^{2} \)
53 \( 1 - 8.57T + 53T^{2} \)
59 \( 1 + 13.4T + 59T^{2} \)
61 \( 1 - 12.7T + 61T^{2} \)
67 \( 1 - 5.10T + 67T^{2} \)
71 \( 1 - 1.65T + 71T^{2} \)
73 \( 1 + 10.3T + 73T^{2} \)
79 \( 1 + 16.2T + 79T^{2} \)
83 \( 1 + 9.35T + 83T^{2} \)
89 \( 1 - 3.10T + 89T^{2} \)
97 \( 1 + 4.55T + 97T^{2} \)
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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

−8.545631826970720253296483961785, −7.27220979164578819313465357989, −6.92635585998282557249227662702, −6.21877437173903329080020564039, −5.57521110486776199541678524994, −5.00022793722222297160912272938, −3.98407109844095702376812415286, −3.16105278521843829131378944880, −1.68321993376562308574504357026, −0.34772825004131021745878878768, 0.34772825004131021745878878768, 1.68321993376562308574504357026, 3.16105278521843829131378944880, 3.98407109844095702376812415286, 5.00022793722222297160912272938, 5.57521110486776199541678524994, 6.21877437173903329080020564039, 6.92635585998282557249227662702, 7.27220979164578819313465357989, 8.545631826970720253296483961785

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