Properties

Label 2-85800-1.1-c1-0-74
Degree $2$
Conductor $85800$
Sign $-1$
Analytic cond. $685.116$
Root an. cond. $26.1747$
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-s + 4·7-s + 9-s − 11-s − 13-s + 2·17-s + 4·19-s + 4·21-s + 4·23-s + 27-s + 6·29-s − 33-s − 10·37-s − 39-s − 10·41-s − 4·43-s + 9·49-s + 2·51-s − 10·53-s + 4·57-s − 12·59-s − 2·61-s + 4·63-s − 8·67-s + 4·69-s − 8·71-s + 2·73-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.51·7-s + 1/3·9-s − 0.301·11-s − 0.277·13-s + 0.485·17-s + 0.917·19-s + 0.872·21-s + 0.834·23-s + 0.192·27-s + 1.11·29-s − 0.174·33-s − 1.64·37-s − 0.160·39-s − 1.56·41-s − 0.609·43-s + 9/7·49-s + 0.280·51-s − 1.37·53-s + 0.529·57-s − 1.56·59-s − 0.256·61-s + 0.503·63-s − 0.977·67-s + 0.481·69-s − 0.949·71-s + 0.234·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 85800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 85800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(85800\)    =    \(2^{3} \cdot 3 \cdot 5^{2} \cdot 11 \cdot 13\)
Sign: $-1$
Analytic conductor: \(685.116\)
Root analytic conductor: \(26.1747\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{85800} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 85800,\ (\ :1/2),\ -1)\)

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 \( 1 \)
3 \( 1 - T \)
5 \( 1 \)
11 \( 1 + T \)
13 \( 1 + T \)
good7 \( 1 - 4 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
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

−14.07585458716458, −13.85340950778924, −13.35668498657162, −12.64061570614340, −12.09143292278254, −11.79495165046882, −11.20379682256527, −10.64977800503360, −10.19922601861682, −9.737604492303445, −8.944002144849675, −8.642689230155592, −8.129417464123340, −7.594252445483105, −7.267605792824229, −6.615845851882909, −5.856165850482684, −5.078829687536189, −4.918172162947348, −4.394131941480071, −3.318780480169980, −3.180817371101257, −2.301013889351919, −1.501163918793157, −1.278685302024058, 0, 1.278685302024058, 1.501163918793157, 2.301013889351919, 3.180817371101257, 3.318780480169980, 4.394131941480071, 4.918172162947348, 5.078829687536189, 5.856165850482684, 6.615845851882909, 7.267605792824229, 7.594252445483105, 8.129417464123340, 8.642689230155592, 8.944002144849675, 9.737604492303445, 10.19922601861682, 10.64977800503360, 11.20379682256527, 11.79495165046882, 12.09143292278254, 12.64061570614340, 13.35668498657162, 13.85340950778924, 14.07585458716458

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