Properties

Label 2-154560-1.1-c1-0-66
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 5-s − 7-s + 9-s + 4·13-s + 15-s + 2·19-s − 21-s − 23-s + 25-s + 27-s − 6·29-s − 2·31-s − 35-s + 10·37-s + 4·39-s + 6·41-s − 4·43-s + 45-s − 6·47-s + 49-s + 6·53-s + 2·57-s + 12·59-s + 10·61-s − 63-s + 4·65-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.447·5-s − 0.377·7-s + 1/3·9-s + 1.10·13-s + 0.258·15-s + 0.458·19-s − 0.218·21-s − 0.208·23-s + 1/5·25-s + 0.192·27-s − 1.11·29-s − 0.359·31-s − 0.169·35-s + 1.64·37-s + 0.640·39-s + 0.937·41-s − 0.609·43-s + 0.149·45-s − 0.875·47-s + 1/7·49-s + 0.824·53-s + 0.264·57-s + 1.56·59-s + 1.28·61-s − 0.125·63-s + 0.496·65-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: $\chi_{154560} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 154560,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.270040218\)
\(L(\frac12)\) \(\approx\) \(4.270040218\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 - 8 T + p T^{2} \)
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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

−13.35956634665767, −12.87001334914578, −12.67883749329966, −11.79307913860414, −11.29557384045620, −11.09641979959068, −10.21284858614106, −9.998754573759118, −9.420484116361194, −9.021712600168147, −8.555788148368428, −7.981249882829842, −7.559793862541555, −6.927565314040800, −6.459266994455567, −5.871572263882324, −5.524990368784235, −4.816582061549780, −4.106793762806525, −3.650479561329907, −3.216298539228505, −2.418996084325432, −2.023818821416571, −1.202146085312142, −0.6227786002808848, 0.6227786002808848, 1.202146085312142, 2.023818821416571, 2.418996084325432, 3.216298539228505, 3.650479561329907, 4.106793762806525, 4.816582061549780, 5.524990368784235, 5.871572263882324, 6.459266994455567, 6.927565314040800, 7.559793862541555, 7.981249882829842, 8.555788148368428, 9.021712600168147, 9.420484116361194, 9.998754573759118, 10.21284858614106, 11.09641979959068, 11.29557384045620, 11.79307913860414, 12.67883749329966, 12.87001334914578, 13.35956634665767

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