Properties

Degree $2$
Conductor $380880$
Sign $-1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5-s − 4·11-s − 2·13-s − 6·17-s + 4·19-s + 25-s + 2·29-s + 2·37-s − 10·41-s − 4·43-s − 7·49-s + 6·53-s − 4·55-s − 4·59-s + 10·61-s − 2·65-s − 12·67-s − 8·71-s + 10·73-s − 8·79-s + 4·83-s − 6·85-s + 18·89-s + 4·95-s − 2·97-s + 101-s + 103-s + ⋯
L(s)  = 1  + 0.447·5-s − 1.20·11-s − 0.554·13-s − 1.45·17-s + 0.917·19-s + 1/5·25-s + 0.371·29-s + 0.328·37-s − 1.56·41-s − 0.609·43-s − 49-s + 0.824·53-s − 0.539·55-s − 0.520·59-s + 1.28·61-s − 0.248·65-s − 1.46·67-s − 0.949·71-s + 1.17·73-s − 0.900·79-s + 0.439·83-s − 0.650·85-s + 1.90·89-s + 0.410·95-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380880\)    =    \(2^{4} \cdot 3^{2} \cdot 5 \cdot 23^{2}\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{380880} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 380880,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
23 \( 1 \)
good7 \( 1 + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 18 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−12.87819765935830, −12.07538906648337, −11.95672562496771, −11.25686823528371, −10.93346843777259, −10.34715596541767, −10.04204400067938, −9.620286936603892, −9.104417439692610, −8.518470504307458, −8.283944328228888, −7.597647918381631, −7.195685444257167, −6.769864692018939, −6.214569605238528, −5.719325586231543, −5.149387315669326, −4.793699814790722, −4.443923334267198, −3.530725885794359, −3.133560044659392, −2.512479052672348, −2.105175135070797, −1.526510788109718, −0.6382363707567199, 0, 0.6382363707567199, 1.526510788109718, 2.105175135070797, 2.512479052672348, 3.133560044659392, 3.530725885794359, 4.443923334267198, 4.793699814790722, 5.149387315669326, 5.719325586231543, 6.214569605238528, 6.769864692018939, 7.195685444257167, 7.597647918381631, 8.283944328228888, 8.518470504307458, 9.104417439692610, 9.620286936603892, 10.04204400067938, 10.34715596541767, 10.93346843777259, 11.25686823528371, 11.95672562496771, 12.07538906648337, 12.87819765935830

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