Properties

Label 2-188760-1.1-c1-0-36
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 + 2·17-s + 4·19-s + 4·21-s + 6·23-s + 25-s + 27-s + 2·29-s − 8·31-s − 4·35-s + 6·37-s − 39-s + 10·41-s − 8·43-s − 45-s + 12·47-s + 9·49-s + 2·51-s + 8·53-s + 4·57-s + 4·59-s + 2·61-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 + 0.485·17-s + 0.917·19-s + 0.872·21-s + 1.25·23-s + 1/5·25-s + 0.192·27-s + 0.371·29-s − 1.43·31-s − 0.676·35-s + 0.986·37-s − 0.160·39-s + 1.56·41-s − 1.21·43-s − 0.149·45-s + 1.75·47-s + 9/7·49-s + 0.280·51-s + 1.09·53-s + 0.529·57-s + 0.520·59-s + 0.256·61-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 188760,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(5.309341757\)
\(L(\frac12)\) \(\approx\) \(5.309341757\)
\(L(\frac{3}{2})\) 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 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 - 8 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 6 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 4 T + p T^{2} \)
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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

−13.15992321665426, −12.60446276448050, −12.16973480834744, −11.68194469528614, −11.17666665447128, −10.92407716811888, −10.36251308075037, −9.686335512172352, −9.299803786189477, −8.683084715430059, −8.443835429358759, −7.681681190681212, −7.512657771878659, −7.166040227738488, −6.402135229388248, −5.562562249581583, −5.247996274985481, −4.788368029824703, −4.093373601135998, −3.782242371550832, −2.966350283966996, −2.522765654587343, −1.849025502620934, −1.135122293832302, −0.7191450143950601, 0.7191450143950601, 1.135122293832302, 1.849025502620934, 2.522765654587343, 2.966350283966996, 3.782242371550832, 4.093373601135998, 4.788368029824703, 5.247996274985481, 5.562562249581583, 6.402135229388248, 7.166040227738488, 7.512657771878659, 7.681681190681212, 8.443835429358759, 8.683084715430059, 9.299803786189477, 9.686335512172352, 10.36251308075037, 10.92407716811888, 11.17666665447128, 11.68194469528614, 12.16973480834744, 12.60446276448050, 13.15992321665426

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