Properties

Label 2-162240-1.1-c1-0-131
Degree $2$
Conductor $162240$
Sign $1$
Analytic cond. $1295.49$
Root an. cond. $35.9929$
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 − 2·7-s + 9-s + 4·11-s − 15-s + 8·17-s − 6·19-s − 2·21-s + 6·23-s + 25-s + 27-s + 4·29-s + 4·33-s + 2·35-s − 2·37-s + 2·41-s + 4·43-s − 45-s − 3·49-s + 8·51-s + 10·53-s − 4·55-s − 6·57-s + 4·59-s + 10·61-s − 2·63-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s − 0.755·7-s + 1/3·9-s + 1.20·11-s − 0.258·15-s + 1.94·17-s − 1.37·19-s − 0.436·21-s + 1.25·23-s + 1/5·25-s + 0.192·27-s + 0.742·29-s + 0.696·33-s + 0.338·35-s − 0.328·37-s + 0.312·41-s + 0.609·43-s − 0.149·45-s − 3/7·49-s + 1.12·51-s + 1.37·53-s − 0.539·55-s − 0.794·57-s + 0.520·59-s + 1.28·61-s − 0.251·63-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 162240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(162240\)    =    \(2^{6} \cdot 3 \cdot 5 \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(1295.49\)
Root analytic conductor: \(35.9929\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{162240} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 162240,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.345864718\)
\(L(\frac12)\) \(\approx\) \(4.345864718\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
13 \( 1 \)
good7 \( 1 + 2 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
17 \( 1 - 8 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 - 4 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 8 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 - 16 T + p T^{2} \)
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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

−13.20372492940880, −12.72461529627507, −12.35998865982542, −12.01817023538600, −11.37660868753693, −10.89446492259464, −10.28388875472039, −9.902586353134127, −9.409547906430433, −8.925477774862690, −8.488463727893202, −8.035234353100351, −7.385228186329249, −6.992493858776540, −6.389038376865144, −6.141236797847208, −5.200556679154195, −4.859270496627467, −3.957198692799185, −3.659521243622884, −3.314401645124402, −2.523254387501152, −1.992277377544650, −0.9019988417601263, −0.7931488456880125, 0.7931488456880125, 0.9019988417601263, 1.992277377544650, 2.523254387501152, 3.314401645124402, 3.659521243622884, 3.957198692799185, 4.859270496627467, 5.200556679154195, 6.141236797847208, 6.389038376865144, 6.992493858776540, 7.385228186329249, 8.035234353100351, 8.488463727893202, 8.925477774862690, 9.409547906430433, 9.902586353134127, 10.28388875472039, 10.89446492259464, 11.37660868753693, 12.01817023538600, 12.35998865982542, 12.72461529627507, 13.20372492940880

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