Properties

Label 2-185900-1.1-c1-0-23
Degree $2$
Conductor $185900$
Sign $-1$
Analytic cond. $1484.41$
Root an. cond. $38.5281$
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  + 2·3-s − 4·7-s + 9-s + 11-s + 4·19-s − 8·21-s + 6·23-s − 4·27-s − 6·29-s − 8·31-s + 2·33-s + 2·37-s − 6·41-s − 8·43-s + 6·47-s + 9·49-s + 6·53-s + 8·57-s + 12·59-s + 2·61-s − 4·63-s − 10·67-s + 12·69-s + 12·71-s − 16·73-s − 4·77-s + 8·79-s + ⋯
L(s)  = 1  + 1.15·3-s − 1.51·7-s + 1/3·9-s + 0.301·11-s + 0.917·19-s − 1.74·21-s + 1.25·23-s − 0.769·27-s − 1.11·29-s − 1.43·31-s + 0.348·33-s + 0.328·37-s − 0.937·41-s − 1.21·43-s + 0.875·47-s + 9/7·49-s + 0.824·53-s + 1.05·57-s + 1.56·59-s + 0.256·61-s − 0.503·63-s − 1.22·67-s + 1.44·69-s + 1.42·71-s − 1.87·73-s − 0.455·77-s + 0.900·79-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 185900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 185900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(185900\)    =    \(2^{2} \cdot 5^{2} \cdot 11 \cdot 13^{2}\)
Sign: $-1$
Analytic conductor: \(1484.41\)
Root analytic conductor: \(38.5281\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 185900,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
13 \( 1 \)
good3 \( 1 - 2 T + p T^{2} \)
7 \( 1 + 4 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 10 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 16 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−13.32708354134553, −13.16726571423253, −12.57161824129951, −12.02065399353313, −11.52629806510504, −11.00305395281972, −10.42857380708933, −9.718958106138216, −9.644304219818891, −9.046763856981296, −8.779021391935911, −8.234052472387562, −7.510796887044312, −7.071324652861625, −6.893855279504118, −6.054926156328935, −5.565414940660799, −5.154283023387349, −4.205932450645074, −3.694959885938043, −3.277349743659492, −2.989024553122660, −2.266737010856106, −1.686078510723295, −0.8038805660167673, 0, 0.8038805660167673, 1.686078510723295, 2.266737010856106, 2.989024553122660, 3.277349743659492, 3.694959885938043, 4.205932450645074, 5.154283023387349, 5.565414940660799, 6.054926156328935, 6.893855279504118, 7.071324652861625, 7.510796887044312, 8.234052472387562, 8.779021391935911, 9.046763856981296, 9.644304219818891, 9.718958106138216, 10.42857380708933, 11.00305395281972, 11.52629806510504, 12.02065399353313, 12.57161824129951, 13.16726571423253, 13.32708354134553

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