Properties

Label 2-33600-1.1-c1-0-42
Degree $2$
Conductor $33600$
Sign $1$
Analytic cond. $268.297$
Root an. cond. $16.3797$
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 + 7-s + 9-s − 2·13-s − 6·17-s − 4·19-s + 21-s + 4·23-s + 27-s − 6·29-s + 6·37-s − 2·39-s + 6·41-s + 4·43-s + 8·47-s + 49-s − 6·51-s + 14·53-s − 4·57-s − 4·59-s + 2·61-s + 63-s − 12·67-s + 4·69-s + 12·71-s − 10·73-s − 8·79-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.377·7-s + 1/3·9-s − 0.554·13-s − 1.45·17-s − 0.917·19-s + 0.218·21-s + 0.834·23-s + 0.192·27-s − 1.11·29-s + 0.986·37-s − 0.320·39-s + 0.937·41-s + 0.609·43-s + 1.16·47-s + 1/7·49-s − 0.840·51-s + 1.92·53-s − 0.529·57-s − 0.520·59-s + 0.256·61-s + 0.125·63-s − 1.46·67-s + 0.481·69-s + 1.42·71-s − 1.17·73-s − 0.900·79-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 33600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(33600\)    =    \(2^{6} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $1$
Analytic conductor: \(268.297\)
Root analytic conductor: \(16.3797\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{33600} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 33600,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.434130917\)
\(L(\frac12)\) \(\approx\) \(2.434130917\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 - T \)
good11 \( 1 + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 14 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
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

−15.03546579775222, −14.59335528567435, −13.96494549976840, −13.34989147073508, −13.00462269852494, −12.52298066972466, −11.77162039909229, −11.25576799952117, −10.70402679045614, −10.31912984761371, −9.417236844753850, −9.023188654068974, −8.703031067097705, −7.880096787078670, −7.416174978099193, −6.912509740538590, −6.206311212064544, −5.568973958910915, −4.791258734257493, −4.235807175303473, −3.818673490257556, −2.619820288909873, −2.453519454815927, −1.586467878564738, −0.5549893645345528, 0.5549893645345528, 1.586467878564738, 2.453519454815927, 2.619820288909873, 3.818673490257556, 4.235807175303473, 4.791258734257493, 5.568973958910915, 6.206311212064544, 6.912509740538590, 7.416174978099193, 7.880096787078670, 8.703031067097705, 9.023188654068974, 9.417236844753850, 10.31912984761371, 10.70402679045614, 11.25576799952117, 11.77162039909229, 12.52298066972466, 13.00462269852494, 13.34989147073508, 13.96494549976840, 14.59335528567435, 15.03546579775222

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