Properties

Label 2-188760-1.1-c1-0-32
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 5-s + 4·7-s + 9-s − 13-s + 15-s − 6·17-s − 4·19-s + 4·21-s + 8·23-s + 25-s + 27-s + 4·35-s + 12·37-s − 39-s + 10·41-s + 4·43-s + 45-s + 6·47-s + 9·49-s − 6·51-s + 6·53-s − 4·57-s − 4·59-s + 6·61-s + 4·63-s − 65-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.447·5-s + 1.51·7-s + 1/3·9-s − 0.277·13-s + 0.258·15-s − 1.45·17-s − 0.917·19-s + 0.872·21-s + 1.66·23-s + 1/5·25-s + 0.192·27-s + 0.676·35-s + 1.97·37-s − 0.160·39-s + 1.56·41-s + 0.609·43-s + 0.149·45-s + 0.875·47-s + 9/7·49-s − 0.840·51-s + 0.824·53-s − 0.529·57-s − 0.520·59-s + 0.768·61-s + 0.503·63-s − 0.124·65-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 188760,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(5.682422122\)
\(L(\frac12)\) \(\approx\) \(5.682422122\)
\(L(\frac{3}{2})\) 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 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 12 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 - 14 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 4 T + p T^{2} \)
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

−13.27184680933690, −12.69113522459114, −12.32906233411092, −11.54337215637138, −11.11056975793526, −10.82484112701081, −10.52360064227738, −9.581617343968429, −9.192163145821601, −8.985183992910900, −8.267737934322789, −7.958469701693121, −7.480831675459749, −6.770907995076066, −6.530451797878380, −5.706603794789206, −5.206071796637991, −4.673770463369687, −4.249591338960918, −3.852372218900036, −2.697268107324562, −2.469222804003325, −2.029520812675119, −1.192353297643562, −0.6986044757992831, 0.6986044757992831, 1.192353297643562, 2.029520812675119, 2.469222804003325, 2.697268107324562, 3.852372218900036, 4.249591338960918, 4.673770463369687, 5.206071796637991, 5.706603794789206, 6.530451797878380, 6.770907995076066, 7.480831675459749, 7.958469701693121, 8.267737934322789, 8.985183992910900, 9.192163145821601, 9.581617343968429, 10.52360064227738, 10.82484112701081, 11.11056975793526, 11.54337215637138, 12.32906233411092, 12.69113522459114, 13.27184680933690

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