Properties

Label 2-369600-1.1-c1-0-122
Degree $2$
Conductor $369600$
Sign $1$
Analytic cond. $2951.27$
Root an. cond. $54.3256$
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 − 11-s + 6·13-s − 6·17-s − 21-s + 27-s + 10·29-s − 33-s + 2·37-s + 6·39-s − 6·41-s − 4·43-s + 49-s − 6·51-s + 2·53-s + 2·61-s − 63-s − 12·67-s + 12·71-s − 6·73-s + 77-s + 4·79-s + 81-s − 4·83-s + 10·87-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.377·7-s + 1/3·9-s − 0.301·11-s + 1.66·13-s − 1.45·17-s − 0.218·21-s + 0.192·27-s + 1.85·29-s − 0.174·33-s + 0.328·37-s + 0.960·39-s − 0.937·41-s − 0.609·43-s + 1/7·49-s − 0.840·51-s + 0.274·53-s + 0.256·61-s − 0.125·63-s − 1.46·67-s + 1.42·71-s − 0.702·73-s + 0.113·77-s + 0.450·79-s + 1/9·81-s − 0.439·83-s + 1.07·87-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 369600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 369600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(369600\)    =    \(2^{6} \cdot 3 \cdot 5^{2} \cdot 7 \cdot 11\)
Sign: $1$
Analytic conductor: \(2951.27\)
Root analytic conductor: \(54.3256\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{369600} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 369600,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.860880690\)
\(L(\frac12)\) \(\approx\) \(2.860880690\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
11 \( 1 + T \)
good13 \( 1 - 6 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 10 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
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

−12.59032985453621, −12.12277264167488, −11.48423942172781, −11.15192944838705, −10.70136356952050, −10.13882602703870, −9.935148912125263, −9.095291752998490, −8.881445511008959, −8.379377709944499, −8.182429226238616, −7.462277536158498, −6.866951298691395, −6.482362618329179, −6.208152091642723, −5.525311662872596, −4.892591088067109, −4.412778489020460, −3.931226236812568, −3.408904754338472, −2.879011573680199, −2.422857009824447, −1.713040051665192, −1.178459287295110, −0.4305170502439346, 0.4305170502439346, 1.178459287295110, 1.713040051665192, 2.422857009824447, 2.879011573680199, 3.408904754338472, 3.931226236812568, 4.412778489020460, 4.892591088067109, 5.525311662872596, 6.208152091642723, 6.482362618329179, 6.866951298691395, 7.462277536158498, 8.182429226238616, 8.379377709944499, 8.881445511008959, 9.095291752998490, 9.935148912125263, 10.13882602703870, 10.70136356952050, 11.15192944838705, 11.48423942172781, 12.12277264167488, 12.59032985453621

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