Properties

Label 2-3800-1.1-c1-0-81
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.41·3-s + 2.82·7-s − 0.999·9-s − 0.828·11-s − 3.41·13-s − 4.82·17-s − 19-s + 4.00·21-s − 4·23-s − 5.65·27-s − 0.828·29-s − 1.17·33-s − 10.2·37-s − 4.82·39-s − 0.828·41-s − 2.82·43-s + 8.48·47-s + 1.00·49-s − 6.82·51-s − 13.0·53-s − 1.41·57-s + 2.82·59-s − 1.65·61-s − 2.82·63-s + 9.41·67-s − 5.65·69-s + 15.3·71-s + ⋯
L(s)  = 1  + 0.816·3-s + 1.06·7-s − 0.333·9-s − 0.249·11-s − 0.946·13-s − 1.17·17-s − 0.229·19-s + 0.872·21-s − 0.834·23-s − 1.08·27-s − 0.153·29-s − 0.203·33-s − 1.68·37-s − 0.773·39-s − 0.129·41-s − 0.431·43-s + 1.23·47-s + 0.142·49-s − 0.956·51-s − 1.79·53-s − 0.187·57-s + 0.368·59-s − 0.212·61-s − 0.356·63-s + 1.15·67-s − 0.681·69-s + 1.81·71-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 3800,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.41T + 3T^{2} \)
7 \( 1 - 2.82T + 7T^{2} \)
11 \( 1 + 0.828T + 11T^{2} \)
13 \( 1 + 3.41T + 13T^{2} \)
17 \( 1 + 4.82T + 17T^{2} \)
23 \( 1 + 4T + 23T^{2} \)
29 \( 1 + 0.828T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + 10.2T + 37T^{2} \)
41 \( 1 + 0.828T + 41T^{2} \)
43 \( 1 + 2.82T + 43T^{2} \)
47 \( 1 - 8.48T + 47T^{2} \)
53 \( 1 + 13.0T + 53T^{2} \)
59 \( 1 - 2.82T + 59T^{2} \)
61 \( 1 + 1.65T + 61T^{2} \)
67 \( 1 - 9.41T + 67T^{2} \)
71 \( 1 - 15.3T + 71T^{2} \)
73 \( 1 + 12.1T + 73T^{2} \)
79 \( 1 + 9.17T + 79T^{2} \)
83 \( 1 + 8T + 83T^{2} \)
89 \( 1 + 3.17T + 89T^{2} \)
97 \( 1 - 2.24T + 97T^{2} \)
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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

−8.269738606703878644712072772452, −7.55194113962969876212176507192, −6.84961150136849014053366910403, −5.80609691758732056345161171967, −4.99932687992669003569647448932, −4.33551937764430817000688339613, −3.35482701674970837480485780563, −2.34836520621442876982072622037, −1.82472000436605057785289224595, 0, 1.82472000436605057785289224595, 2.34836520621442876982072622037, 3.35482701674970837480485780563, 4.33551937764430817000688339613, 4.99932687992669003569647448932, 5.80609691758732056345161171967, 6.84961150136849014053366910403, 7.55194113962969876212176507192, 8.269738606703878644712072772452

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