Properties

Label 2-31200-1.1-c1-0-0
Degree $2$
Conductor $31200$
Sign $1$
Analytic cond. $249.133$
Root an. cond. $15.7839$
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 + 2·7-s + 9-s − 4·11-s − 13-s + 6·19-s − 2·21-s − 6·23-s − 27-s − 8·29-s + 4·33-s − 10·37-s + 39-s − 10·41-s − 12·43-s − 12·47-s − 3·49-s + 6·53-s − 6·57-s − 12·59-s − 10·61-s + 2·63-s − 4·67-s + 6·69-s + 8·71-s + 4·73-s − 8·77-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.755·7-s + 1/3·9-s − 1.20·11-s − 0.277·13-s + 1.37·19-s − 0.436·21-s − 1.25·23-s − 0.192·27-s − 1.48·29-s + 0.696·33-s − 1.64·37-s + 0.160·39-s − 1.56·41-s − 1.82·43-s − 1.75·47-s − 3/7·49-s + 0.824·53-s − 0.794·57-s − 1.56·59-s − 1.28·61-s + 0.251·63-s − 0.488·67-s + 0.722·69-s + 0.949·71-s + 0.468·73-s − 0.911·77-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 31200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(31200\)    =    \(2^{5} \cdot 3 \cdot 5^{2} \cdot 13\)
Sign: $1$
Analytic conductor: \(249.133\)
Root analytic conductor: \(15.7839\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 31200,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7084082634\)
\(L(\frac12)\) \(\approx\) \(0.7084082634\)
\(L(\frac{3}{2})\) 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 \( 1 \)
13 \( 1 + T \)
good7 \( 1 - 2 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 - 4 T + p T^{2} \)
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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

−15.26546571676621, −14.57658120378603, −13.96662359571269, −13.45328138030650, −13.10025623075258, −12.18421669593137, −11.96693794765038, −11.39544454458162, −10.86247393208148, −10.22661371275775, −9.912464096362807, −9.231065786834115, −8.405621384812758, −7.913321797015951, −7.520765434323336, −6.849713742736701, −6.156664542913421, −5.361630798865747, −5.121680255216507, −4.622804054630586, −3.516953586466659, −3.181370897855001, −1.844389963874960, −1.743638282440387, −0.3172624167610356, 0.3172624167610356, 1.743638282440387, 1.844389963874960, 3.181370897855001, 3.516953586466659, 4.622804054630586, 5.121680255216507, 5.361630798865747, 6.156664542913421, 6.849713742736701, 7.520765434323336, 7.913321797015951, 8.405621384812758, 9.231065786834115, 9.912464096362807, 10.22661371275775, 10.86247393208148, 11.39544454458162, 11.96693794765038, 12.18421669593137, 13.10025623075258, 13.45328138030650, 13.96662359571269, 14.57658120378603, 15.26546571676621

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