Properties

Label 2-203280-1.1-c1-0-15
Degree $2$
Conductor $203280$
Sign $1$
Analytic cond. $1623.19$
Root an. cond. $40.2889$
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 + 2·13-s − 15-s − 6·17-s − 4·19-s + 21-s − 4·23-s + 25-s + 27-s − 6·29-s − 35-s + 6·37-s + 2·39-s − 6·41-s − 4·43-s − 45-s − 8·47-s + 49-s − 6·51-s + 14·53-s − 4·57-s + 4·59-s + 2·61-s + 63-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s + 0.377·7-s + 1/3·9-s + 0.554·13-s − 0.258·15-s − 1.45·17-s − 0.917·19-s + 0.218·21-s − 0.834·23-s + 1/5·25-s + 0.192·27-s − 1.11·29-s − 0.169·35-s + 0.986·37-s + 0.320·39-s − 0.937·41-s − 0.609·43-s − 0.149·45-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 + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 203280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 203280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(203280\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 7 \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(1623.19\)
Root analytic conductor: \(40.2889\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 203280,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.472929163\)
\(L(\frac12)\) \(\approx\) \(1.472929163\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
11 \( 1 \)
good13 \( 1 - 2 T + p T^{2} \)
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

−13.06645100734994, −12.77287144412228, −12.04987796315889, −11.43420473050811, −11.40895883507533, −10.64695911589432, −10.32008947056141, −9.732351870680522, −9.092202240452328, −8.769321136286607, −8.278788032563396, −7.986807924290233, −7.328552712648398, −6.837639609772706, −6.393850818343900, −5.829381513818968, −5.179295183108379, −4.515051454909583, −4.170907644948112, −3.692274322004057, −3.111160689840339, −2.227320023136082, −2.055107601971347, −1.269114313004797, −0.3223185429015088, 0.3223185429015088, 1.269114313004797, 2.055107601971347, 2.227320023136082, 3.111160689840339, 3.692274322004057, 4.170907644948112, 4.515051454909583, 5.179295183108379, 5.829381513818968, 6.393850818343900, 6.837639609772706, 7.328552712648398, 7.986807924290233, 8.278788032563396, 8.769321136286607, 9.092202240452328, 9.732351870680522, 10.32008947056141, 10.64695911589432, 11.40895883507533, 11.43420473050811, 12.04987796315889, 12.77287144412228, 13.06645100734994

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