Properties

Label 2-152880-1.1-c1-0-46
Degree $2$
Conductor $152880$
Sign $1$
Analytic cond. $1220.75$
Root an. cond. $34.9392$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 5-s + 9-s − 4·11-s + 13-s − 15-s − 6·17-s + 8·19-s − 4·23-s + 25-s − 27-s − 2·29-s + 4·31-s + 4·33-s + 10·37-s − 39-s − 6·41-s − 8·43-s + 45-s + 8·47-s + 6·51-s − 2·53-s − 4·55-s − 8·57-s − 8·59-s + 2·61-s + 65-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.447·5-s + 1/3·9-s − 1.20·11-s + 0.277·13-s − 0.258·15-s − 1.45·17-s + 1.83·19-s − 0.834·23-s + 1/5·25-s − 0.192·27-s − 0.371·29-s + 0.718·31-s + 0.696·33-s + 1.64·37-s − 0.160·39-s − 0.937·41-s − 1.21·43-s + 0.149·45-s + 1.16·47-s + 0.840·51-s − 0.274·53-s − 0.539·55-s − 1.05·57-s − 1.04·59-s + 0.256·61-s + 0.124·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 152880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 152880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(152880\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 7^{2} \cdot 13\)
Sign: $1$
Analytic conductor: \(1220.75\)
Root analytic conductor: \(34.9392\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{152880} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 152880,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.783724032\)
\(L(\frac12)\) \(\approx\) \(1.783724032\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
7 \( 1 \)
13 \( 1 - T \)
good11 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 18 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−13.43393453075604, −12.95285369690332, −12.27678012011827, −11.93010766631379, −11.31884589445999, −11.02786370267336, −10.43781285441713, −10.04681829506308, −9.507233952462686, −9.173358714004592, −8.370930986273367, −7.993595564216868, −7.432928125116166, −6.980299629044200, −6.212902627022794, −6.042876326689095, −5.368553082359104, −4.792384313980657, −4.610817946604804, −3.606639553271793, −3.196618194581615, −2.306601009792963, −2.050804592021997, −1.061834327704170, −0.4529486176202810, 0.4529486176202810, 1.061834327704170, 2.050804592021997, 2.306601009792963, 3.196618194581615, 3.606639553271793, 4.610817946604804, 4.792384313980657, 5.368553082359104, 6.042876326689095, 6.212902627022794, 6.980299629044200, 7.432928125116166, 7.993595564216868, 8.370930986273367, 9.173358714004592, 9.507233952462686, 10.04681829506308, 10.43781285441713, 11.02786370267336, 11.31884589445999, 11.93010766631379, 12.27678012011827, 12.95285369690332, 13.43393453075604

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