Properties

Degree $2$
Conductor $235200$
Sign $-1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 9-s − 5·11-s + 13-s + 2·17-s + 7·19-s + 3·23-s − 27-s + 6·31-s + 5·33-s − 5·37-s − 39-s − 9·41-s − 10·43-s − 13·47-s − 2·51-s − 53-s − 7·57-s + 4·59-s + 2·61-s − 6·67-s − 3·69-s + 2·71-s − 4·73-s + 14·79-s + 81-s + 10·83-s + ⋯
L(s)  = 1  − 0.577·3-s + 1/3·9-s − 1.50·11-s + 0.277·13-s + 0.485·17-s + 1.60·19-s + 0.625·23-s − 0.192·27-s + 1.07·31-s + 0.870·33-s − 0.821·37-s − 0.160·39-s − 1.40·41-s − 1.52·43-s − 1.89·47-s − 0.280·51-s − 0.137·53-s − 0.927·57-s + 0.520·59-s + 0.256·61-s − 0.733·67-s − 0.361·69-s + 0.237·71-s − 0.468·73-s + 1.57·79-s + 1/9·81-s + 1.09·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 235200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 235200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(235200\)    =    \(2^{6} \cdot 3 \cdot 5^{2} \cdot 7^{2}\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{235200} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 235200,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
5 \( 1 \)
7 \( 1 \)
good11 \( 1 + 5 T + p T^{2} \)
13 \( 1 - T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 7 T + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 6 T + p T^{2} \)
37 \( 1 + 5 T + p T^{2} \)
41 \( 1 + 9 T + p T^{2} \)
43 \( 1 + 10 T + p T^{2} \)
47 \( 1 + 13 T + p T^{2} \)
53 \( 1 + T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 6 T + p T^{2} \)
71 \( 1 - 2 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 - 14 T + p T^{2} \)
83 \( 1 - 10 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 + 8 T + p T^{2} \)
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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

−13.32820158616644, −12.65414723843876, −12.08197694743934, −11.78972982530659, −11.36611731324043, −10.77777359139160, −10.34230864469389, −9.968917584349113, −9.612628178582380, −8.932134980710776, −8.264255111694507, −7.991664136223690, −7.532734547366449, −6.802191955460905, −6.635490131148321, −5.849700010588014, −5.281465212456156, −5.054039654343743, −4.710888980257086, −3.710497397942294, −3.209983686641418, −2.917414158102332, −2.006810873823467, −1.421860845994185, −0.7262699902245839, 0, 0.7262699902245839, 1.421860845994185, 2.006810873823467, 2.917414158102332, 3.209983686641418, 3.710497397942294, 4.710888980257086, 5.054039654343743, 5.281465212456156, 5.849700010588014, 6.635490131148321, 6.802191955460905, 7.532734547366449, 7.991664136223690, 8.264255111694507, 8.932134980710776, 9.612628178582380, 9.968917584349113, 10.34230864469389, 10.77777359139160, 11.36611731324043, 11.78972982530659, 12.08197694743934, 12.65414723843876, 13.32820158616644

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