Properties

Label 2-440e2-1.1-c1-0-85
Degree $2$
Conductor $193600$
Sign $-1$
Analytic cond. $1545.90$
Root an. cond. $39.3179$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·3-s − 7-s + 6·9-s − 5·17-s − 7·19-s + 3·21-s + 8·23-s − 9·27-s + 3·29-s − 5·31-s − 37-s + 8·41-s − 10·43-s − 6·49-s + 15·51-s − 53-s + 21·57-s − 12·59-s + 5·61-s − 6·63-s − 4·67-s − 24·69-s − 7·71-s + 2·73-s + 4·79-s + 9·81-s − 9·87-s + ⋯
L(s)  = 1  − 1.73·3-s − 0.377·7-s + 2·9-s − 1.21·17-s − 1.60·19-s + 0.654·21-s + 1.66·23-s − 1.73·27-s + 0.557·29-s − 0.898·31-s − 0.164·37-s + 1.24·41-s − 1.52·43-s − 6/7·49-s + 2.10·51-s − 0.137·53-s + 2.78·57-s − 1.56·59-s + 0.640·61-s − 0.755·63-s − 0.488·67-s − 2.88·69-s − 0.830·71-s + 0.234·73-s + 0.450·79-s + 81-s − 0.964·87-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 193600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 193600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(193600\)    =    \(2^{6} \cdot 5^{2} \cdot 11^{2}\)
Sign: $-1$
Analytic conductor: \(1545.90\)
Root analytic conductor: \(39.3179\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 193600,\ (\ :1/2),\ -1)\)

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 \( 1 \)
11 \( 1 \)
good3 \( 1 + p T + p T^{2} \)
7 \( 1 + T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 + 5 T + p T^{2} \)
19 \( 1 + 7 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 + 5 T + p T^{2} \)
37 \( 1 + T + p T^{2} \)
41 \( 1 - 8 T + p T^{2} \)
43 \( 1 + 10 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 5 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 7 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + 7 T + p T^{2} \)
97 \( 1 + 8 T + p T^{2} \)
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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

−13.07471686670859, −12.71845279129036, −12.56201596723320, −11.80191774259992, −11.36020169361586, −11.02029686495269, −10.64455243093541, −10.29352911805247, −9.637560278281807, −9.077974811640766, −8.726334463024103, −8.038099776932004, −7.322530033188773, −6.839158617609418, −6.534606425440800, −6.152319545584522, −5.582684075999644, −4.980271495399731, −4.581236525407060, −4.225341606982331, −3.424088250548849, −2.751657210770956, −1.960438086114676, −1.396063261941988, −0.5445689813891580, 0, 0.5445689813891580, 1.396063261941988, 1.960438086114676, 2.751657210770956, 3.424088250548849, 4.225341606982331, 4.581236525407060, 4.980271495399731, 5.582684075999644, 6.152319545584522, 6.534606425440800, 6.839158617609418, 7.322530033188773, 8.038099776932004, 8.726334463024103, 9.077974811640766, 9.637560278281807, 10.29352911805247, 10.64455243093541, 11.02029686495269, 11.36020169361586, 11.80191774259992, 12.56201596723320, 12.71845279129036, 13.07471686670859

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