Properties

Label 2-188760-1.1-c1-0-21
Degree $2$
Conductor $188760$
Sign $-1$
Analytic cond. $1507.25$
Root an. cond. $38.8233$
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 − 5-s − 2·7-s + 9-s − 13-s + 15-s + 3·17-s − 4·19-s + 2·21-s + 23-s + 25-s − 27-s − 8·29-s − 5·31-s + 2·35-s + 10·37-s + 39-s − 12·41-s − 12·43-s − 45-s − 7·47-s − 3·49-s − 3·51-s − 9·53-s + 4·57-s − 10·59-s − 3·61-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s − 0.755·7-s + 1/3·9-s − 0.277·13-s + 0.258·15-s + 0.727·17-s − 0.917·19-s + 0.436·21-s + 0.208·23-s + 1/5·25-s − 0.192·27-s − 1.48·29-s − 0.898·31-s + 0.338·35-s + 1.64·37-s + 0.160·39-s − 1.87·41-s − 1.82·43-s − 0.149·45-s − 1.02·47-s − 3/7·49-s − 0.420·51-s − 1.23·53-s + 0.529·57-s − 1.30·59-s − 0.384·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 188760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 188760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(188760\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 11^{2} \cdot 13\)
Sign: $-1$
Analytic conductor: \(1507.25\)
Root analytic conductor: \(38.8233\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{188760} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 188760,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
11 \( 1 \)
13 \( 1 + T \)
good7 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - T + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 + 5 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 12 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 + 7 T + p T^{2} \)
53 \( 1 + 9 T + p T^{2} \)
59 \( 1 + 10 T + p T^{2} \)
61 \( 1 + 3 T + p T^{2} \)
67 \( 1 - 6 T + p T^{2} \)
71 \( 1 + 4 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 - T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 - 12 T + p T^{2} \)
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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

−13.18577607814021, −12.89760554396170, −12.39812561432582, −11.98930409128546, −11.43796390687831, −11.04337367546428, −10.63773865033879, −10.00910812123395, −9.511127189802896, −9.374111561093302, −8.490982578000024, −8.018833804349876, −7.667037778762251, −6.916542366855502, −6.634758005886947, −6.135902067254286, −5.559340170602566, −4.989248442159716, −4.595990887237847, −3.844070087679850, −3.357970758133127, −3.013439071368357, −1.934011071478095, −1.623472305632871, −0.5356420041522263, 0, 0.5356420041522263, 1.623472305632871, 1.934011071478095, 3.013439071368357, 3.357970758133127, 3.844070087679850, 4.595990887237847, 4.989248442159716, 5.559340170602566, 6.135902067254286, 6.634758005886947, 6.916542366855502, 7.667037778762251, 8.018833804349876, 8.490982578000024, 9.374111561093302, 9.511127189802896, 10.00910812123395, 10.63773865033879, 11.04337367546428, 11.43796390687831, 11.98930409128546, 12.39812561432582, 12.89760554396170, 13.18577607814021

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