Properties

Label 2-154560-1.1-c1-0-208
Degree $2$
Conductor $154560$
Sign $1$
Analytic cond. $1234.16$
Root an. cond. $35.1307$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $2$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 5-s − 7-s + 9-s − 4·11-s − 2·13-s − 15-s − 2·17-s − 4·19-s − 21-s + 23-s + 25-s + 27-s − 2·29-s − 4·33-s + 35-s − 6·37-s − 2·39-s − 6·41-s − 8·43-s − 45-s − 4·47-s + 49-s − 2·51-s − 2·53-s + 4·55-s − 4·57-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s − 0.377·7-s + 1/3·9-s − 1.20·11-s − 0.554·13-s − 0.258·15-s − 0.485·17-s − 0.917·19-s − 0.218·21-s + 0.208·23-s + 1/5·25-s + 0.192·27-s − 0.371·29-s − 0.696·33-s + 0.169·35-s − 0.986·37-s − 0.320·39-s − 0.937·41-s − 1.21·43-s − 0.149·45-s − 0.583·47-s + 1/7·49-s − 0.280·51-s − 0.274·53-s + 0.539·55-s − 0.529·57-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 154560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 154560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(154560\)    =    \(2^{6} \cdot 3 \cdot 5 \cdot 7 \cdot 23\)
Sign: $1$
Analytic conductor: \(1234.16\)
Root analytic conductor: \(35.1307\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((2,\ 154560,\ (\ :1/2),\ 1)\)

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 \( 1 + T \)
7 \( 1 + T \)
23 \( 1 - T \)
good11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 8 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
97 \( 1 + 18 T + p T^{2} \)
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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

−13.66798944460280, −13.18543500740142, −13.02058409919808, −12.39905373601147, −12.03681965235792, −11.26278949256080, −11.02544300325395, −10.23474386495665, −10.07551348393111, −9.556539823511811, −8.772401279570631, −8.460157471141318, −8.146964245303965, −7.417564187886179, −7.020018152951262, −6.649391009106429, −5.887749546908241, −5.247539793358231, −4.827699175144347, −4.274868865068841, −3.553680753078629, −3.208974622467966, −2.431201053583410, −2.128964427268762, −1.263228026015961, 0, 0, 1.263228026015961, 2.128964427268762, 2.431201053583410, 3.208974622467966, 3.553680753078629, 4.274868865068841, 4.827699175144347, 5.247539793358231, 5.887749546908241, 6.649391009106429, 7.020018152951262, 7.417564187886179, 8.146964245303965, 8.460157471141318, 8.772401279570631, 9.556539823511811, 10.07551348393111, 10.23474386495665, 11.02544300325395, 11.26278949256080, 12.03681965235792, 12.39905373601147, 13.02058409919808, 13.18543500740142, 13.66798944460280

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