Properties

Label 2-31200-1.1-c1-0-13
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 + 4·7-s + 9-s + 2·11-s + 13-s − 2·19-s − 4·21-s − 2·23-s − 27-s + 10·29-s − 4·31-s − 2·33-s − 6·37-s − 39-s − 6·41-s + 8·43-s + 12·47-s + 9·49-s − 14·53-s + 2·57-s − 6·59-s + 2·61-s + 4·63-s − 4·67-s + 2·69-s + 14·73-s + 8·77-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.51·7-s + 1/3·9-s + 0.603·11-s + 0.277·13-s − 0.458·19-s − 0.872·21-s − 0.417·23-s − 0.192·27-s + 1.85·29-s − 0.718·31-s − 0.348·33-s − 0.986·37-s − 0.160·39-s − 0.937·41-s + 1.21·43-s + 1.75·47-s + 9/7·49-s − 1.92·53-s + 0.264·57-s − 0.781·59-s + 0.256·61-s + 0.503·63-s − 0.488·67-s + 0.240·69-s + 1.63·73-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: $\chi_{31200} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 31200,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.487242099\)
\(L(\frac12)\) \(\approx\) \(2.487242099\)
\(L(\frac{3}{2})\) 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 \( 1 - 4 T + p T^{2} \)
11 \( 1 - 2 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 + 2 T + p T^{2} \)
29 \( 1 - 10 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 + 14 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−15.14404279270847, −14.41658719342965, −14.05126990949393, −13.78002399177878, −12.76913996895832, −12.35292715025491, −11.85595555803188, −11.39550769047927, −10.78577725368913, −10.51588370151393, −9.817787116188416, −8.863863557569679, −8.759897947741988, −7.850968586425345, −7.582745874571350, −6.644045667555324, −6.319878471031639, −5.485195468831794, −5.011714888370237, −4.381296969870324, −3.930049411498191, −2.958650429965261, −1.991619495603084, −1.476788799446027, −0.6564176196834244, 0.6564176196834244, 1.476788799446027, 1.991619495603084, 2.958650429965261, 3.930049411498191, 4.381296969870324, 5.011714888370237, 5.485195468831794, 6.319878471031639, 6.644045667555324, 7.582745874571350, 7.850968586425345, 8.759897947741988, 8.863863557569679, 9.817787116188416, 10.51588370151393, 10.78577725368913, 11.39550769047927, 11.85595555803188, 12.35292715025491, 12.76913996895832, 13.78002399177878, 14.05126990949393, 14.41658719342965, 15.14404279270847

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