Properties

Label 2-7200-1.1-c1-0-11
Degree $2$
Conductor $7200$
Sign $1$
Analytic cond. $57.4922$
Root an. cond. $7.58236$
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  − 4·7-s + 4·11-s − 6·13-s + 2·17-s − 4·19-s − 10·29-s + 4·31-s + 10·37-s − 2·41-s − 4·43-s − 8·47-s + 9·49-s + 2·53-s + 12·59-s − 10·61-s + 12·67-s − 10·73-s − 16·77-s + 4·79-s − 4·83-s + 6·89-s + 24·91-s + 14·97-s − 2·101-s + 4·103-s + 12·107-s − 2·109-s + ⋯
L(s)  = 1  − 1.51·7-s + 1.20·11-s − 1.66·13-s + 0.485·17-s − 0.917·19-s − 1.85·29-s + 0.718·31-s + 1.64·37-s − 0.312·41-s − 0.609·43-s − 1.16·47-s + 9/7·49-s + 0.274·53-s + 1.56·59-s − 1.28·61-s + 1.46·67-s − 1.17·73-s − 1.82·77-s + 0.450·79-s − 0.439·83-s + 0.635·89-s + 2.51·91-s + 1.42·97-s − 0.199·101-s + 0.394·103-s + 1.16·107-s − 0.191·109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(7200\)    =    \(2^{5} \cdot 3^{2} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(57.4922\)
Root analytic conductor: \(7.58236\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{7200} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 7200,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.081313375\)
\(L(\frac12)\) \(\approx\) \(1.081313375\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 \)
good7 \( 1 + 4 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 10 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
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

−7.78145398812000971262690871554, −7.14375287833410078479431736650, −6.50633388063324819338747364862, −6.03330214742327009150557463484, −5.09910422861743261288259915465, −4.22963585877726431047642501901, −3.56169383752156022497502286286, −2.77254402767806087386047253538, −1.89785254017853770558179110971, −0.50389200920024116772639675277, 0.50389200920024116772639675277, 1.89785254017853770558179110971, 2.77254402767806087386047253538, 3.56169383752156022497502286286, 4.22963585877726431047642501901, 5.09910422861743261288259915465, 6.03330214742327009150557463484, 6.50633388063324819338747364862, 7.14375287833410078479431736650, 7.78145398812000971262690871554

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