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Properties

Label 2-243600-1.1-c1-0-23
Degree $2$
Conductor $243600$
Sign $1$
Analytic cond. $1945.15$
Root an. cond. $44.1039$
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 − 4·11-s + 2·13-s − 2·17-s + 4·19-s − 21-s − 27-s + 29-s + 8·31-s + 4·33-s + 10·37-s − 2·39-s − 6·41-s + 12·43-s − 8·47-s + 49-s + 2·51-s − 6·53-s − 4·57-s − 12·59-s − 10·61-s + 63-s − 12·67-s + 16·71-s − 2·73-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.377·7-s + 1/3·9-s − 1.20·11-s + 0.554·13-s − 0.485·17-s + 0.917·19-s − 0.218·21-s − 0.192·27-s + 0.185·29-s + 1.43·31-s + 0.696·33-s + 1.64·37-s − 0.320·39-s − 0.937·41-s + 1.82·43-s − 1.16·47-s + 1/7·49-s + 0.280·51-s − 0.824·53-s − 0.529·57-s − 1.56·59-s − 1.28·61-s + 0.125·63-s − 1.46·67-s + 1.89·71-s − 0.234·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 243600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 243600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(243600\)    =    \(2^{4} \cdot 3 \cdot 5^{2} \cdot 7 \cdot 29\)
Sign: $1$
Analytic conductor: \(1945.15\)
Root analytic conductor: \(44.1039\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 243600,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.606184318\)
\(L(\frac12)\) \(\approx\) \(1.606184318\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
29 \( 1 - T \)
good11 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 16 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 6 T + p T^{2} \)
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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

−12.89869962373426, −12.22672161149556, −12.09394530962497, −11.36922228023010, −10.98830358154630, −10.73345106692241, −10.19029422092064, −9.549950598777250, −9.361981677085610, −8.559315868630450, −8.024997995187550, −7.807098891461228, −7.269413116572174, −6.588964186760643, −6.120208369341181, −5.784137059213853, −5.083311143140075, −4.706052950150197, −4.339096406817738, −3.544394156405611, −2.861997478001702, −2.554144748802066, −1.641151094916993, −1.122572495193215, −0.3920222621861195, 0.3920222621861195, 1.122572495193215, 1.641151094916993, 2.554144748802066, 2.861997478001702, 3.544394156405611, 4.339096406817738, 4.706052950150197, 5.083311143140075, 5.784137059213853, 6.120208369341181, 6.588964186760643, 7.269413116572174, 7.807098891461228, 8.024997995187550, 8.559315868630450, 9.361981677085610, 9.549950598777250, 10.19029422092064, 10.73345106692241, 10.98830358154630, 11.36922228023010, 12.09394530962497, 12.22672161149556, 12.89869962373426

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