Properties

Label 2-138600-1.1-c1-0-0
Degree $2$
Conductor $138600$
Sign $1$
Analytic cond. $1106.72$
Root an. cond. $33.2675$
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  − 7-s − 11-s + 6·13-s − 2·17-s − 8·19-s − 4·23-s − 2·29-s − 8·31-s − 6·37-s + 2·41-s − 8·43-s − 4·47-s + 49-s + 2·53-s + 12·59-s + 10·61-s + 12·67-s + 12·71-s − 10·73-s + 77-s − 8·79-s + 12·83-s − 10·89-s − 6·91-s + 14·97-s + 101-s + 103-s + ⋯
L(s)  = 1  − 0.377·7-s − 0.301·11-s + 1.66·13-s − 0.485·17-s − 1.83·19-s − 0.834·23-s − 0.371·29-s − 1.43·31-s − 0.986·37-s + 0.312·41-s − 1.21·43-s − 0.583·47-s + 1/7·49-s + 0.274·53-s + 1.56·59-s + 1.28·61-s + 1.46·67-s + 1.42·71-s − 1.17·73-s + 0.113·77-s − 0.900·79-s + 1.31·83-s − 1.05·89-s − 0.628·91-s + 1.42·97-s + 0.0995·101-s + 0.0985·103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 138600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 138600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(138600\)    =    \(2^{3} \cdot 3^{2} \cdot 5^{2} \cdot 7 \cdot 11\)
Sign: $1$
Analytic conductor: \(1106.72\)
Root analytic conductor: \(33.2675\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{138600} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 138600,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8940799428\)
\(L(\frac12)\) \(\approx\) \(0.8940799428\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 + T \)
11 \( 1 + T \)
good13 \( 1 - 6 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
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 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
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

−13.24221001214943, −12.90825335634260, −12.79106992398113, −11.91714314595938, −11.41670335412280, −11.03031631750832, −10.55332767777515, −10.14802962590542, −9.589322680482580, −8.838502101462032, −8.614017425818952, −8.229965124947744, −7.571493970713981, −6.770551992565542, −6.595937307963704, −6.022294197673108, −5.469653890781182, −4.963642222903508, −4.039295019769518, −3.831483175024928, −3.363987204351324, −2.318586318790581, −2.038563968992904, −1.263912401519860, −0.2807054265489530, 0.2807054265489530, 1.263912401519860, 2.038563968992904, 2.318586318790581, 3.363987204351324, 3.831483175024928, 4.039295019769518, 4.963642222903508, 5.469653890781182, 6.022294197673108, 6.595937307963704, 6.770551992565542, 7.571493970713981, 8.229965124947744, 8.614017425818952, 8.838502101462032, 9.589322680482580, 10.14802962590542, 10.55332767777515, 11.03031631750832, 11.41670335412280, 11.91714314595938, 12.79106992398113, 12.90825335634260, 13.24221001214943

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