Properties

Label 2-216384-1.1-c1-0-7
Degree $2$
Conductor $216384$
Sign $1$
Analytic cond. $1727.83$
Root an. cond. $41.5672$
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 + 4·5-s + 9-s − 2·11-s + 13-s − 4·15-s − 6·17-s + 19-s − 23-s + 11·25-s − 27-s − 10·29-s − 7·31-s + 2·33-s + 3·37-s − 39-s − 12·41-s + 7·43-s + 4·45-s − 10·47-s + 6·51-s − 12·53-s − 8·55-s − 57-s − 14·59-s − 10·61-s + 4·65-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.78·5-s + 1/3·9-s − 0.603·11-s + 0.277·13-s − 1.03·15-s − 1.45·17-s + 0.229·19-s − 0.208·23-s + 11/5·25-s − 0.192·27-s − 1.85·29-s − 1.25·31-s + 0.348·33-s + 0.493·37-s − 0.160·39-s − 1.87·41-s + 1.06·43-s + 0.596·45-s − 1.45·47-s + 0.840·51-s − 1.64·53-s − 1.07·55-s − 0.132·57-s − 1.82·59-s − 1.28·61-s + 0.496·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 216384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(216384\)    =    \(2^{6} \cdot 3 \cdot 7^{2} \cdot 23\)
Sign: $1$
Analytic conductor: \(1727.83\)
Root analytic conductor: \(41.5672\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{216384} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 216384,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7797123856\)
\(L(\frac12)\) \(\approx\) \(0.7797123856\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
7 \( 1 \)
23 \( 1 + T \)
good5 \( 1 - 4 T + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
13 \( 1 - T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - T + p T^{2} \)
29 \( 1 + 10 T + p T^{2} \)
31 \( 1 + 7 T + p T^{2} \)
37 \( 1 - 3 T + p T^{2} \)
41 \( 1 + 12 T + p T^{2} \)
43 \( 1 - 7 T + p T^{2} \)
47 \( 1 + 10 T + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 + 14 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 5 T + p T^{2} \)
71 \( 1 + 2 T + p T^{2} \)
73 \( 1 + 9 T + p T^{2} \)
79 \( 1 + 9 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
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

−13.11192238181509, −12.71682117361726, −12.14332027559088, −11.37341958864629, −11.05585369077896, −10.69104587191937, −10.25665957251429, −9.616216108511126, −9.316446873609132, −9.013380449444968, −8.312914606428130, −7.684295725972360, −7.133302078547225, −6.644526177659936, −6.096168909779826, −5.850831309287565, −5.320406331678279, −4.808485034109311, −4.404651911974681, −3.480082871598317, −2.989301436011019, −2.217112325238917, −1.716831749141524, −1.488542824231972, −0.2315158910432452, 0.2315158910432452, 1.488542824231972, 1.716831749141524, 2.217112325238917, 2.989301436011019, 3.480082871598317, 4.404651911974681, 4.808485034109311, 5.320406331678279, 5.850831309287565, 6.096168909779826, 6.644526177659936, 7.133302078547225, 7.684295725972360, 8.312914606428130, 9.013380449444968, 9.316446873609132, 9.616216108511126, 10.25665957251429, 10.69104587191937, 11.05585369077896, 11.37341958864629, 12.14332027559088, 12.71682117361726, 13.11192238181509

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