Properties

Degree $2$
Conductor $61200$
Sign $-1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·7-s + 6·11-s − 2·13-s − 17-s + 4·19-s + 4·31-s + 4·37-s − 6·41-s + 8·43-s + 9·49-s − 6·53-s − 4·61-s + 8·67-s − 2·73-s − 24·77-s − 8·79-s + 6·89-s + 8·91-s − 14·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + 4·119-s + ⋯
L(s)  = 1  − 1.51·7-s + 1.80·11-s − 0.554·13-s − 0.242·17-s + 0.917·19-s + 0.718·31-s + 0.657·37-s − 0.937·41-s + 1.21·43-s + 9/7·49-s − 0.824·53-s − 0.512·61-s + 0.977·67-s − 0.234·73-s − 2.73·77-s − 0.900·79-s + 0.635·89-s + 0.838·91-s − 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + 0.366·119-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 61200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 61200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(61200\)    =    \(2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 17\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{61200} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 61200,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 \)
17 \( 1 + T \)
good7 \( 1 + 4 T + p T^{2} \)
11 \( 1 - 6 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 4 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + p T^{2} \)
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

−14.58335055850599, −13.94159065542931, −13.61154636023475, −13.03360779735319, −12.34392613885388, −12.19078262435869, −11.58750687924671, −11.07227207046382, −10.35051021860312, −9.744194209862240, −9.443806593457778, −9.155853347825521, −8.418631644201157, −7.746249447264225, −7.038287719222839, −6.683847821196369, −6.271865359874502, −5.674503199830726, −4.961366223072104, −4.105357296064011, −3.862597999467315, −3.018441364868512, −2.670048502613930, −1.591052018244444, −0.9305271855260058, 0, 0.9305271855260058, 1.591052018244444, 2.670048502613930, 3.018441364868512, 3.862597999467315, 4.105357296064011, 4.961366223072104, 5.674503199830726, 6.271865359874502, 6.683847821196369, 7.038287719222839, 7.746249447264225, 8.418631644201157, 9.155853347825521, 9.443806593457778, 9.744194209862240, 10.35051021860312, 11.07227207046382, 11.58750687924671, 12.19078262435869, 12.34392613885388, 13.03360779735319, 13.61154636023475, 13.94159065542931, 14.58335055850599

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