Properties

Label 2-31200-1.1-c1-0-27
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 4·7-s + 9-s + 4·11-s + 13-s − 2·17-s − 4·19-s + 4·21-s − 27-s + 6·29-s − 4·33-s − 10·37-s − 39-s + 2·41-s + 4·43-s − 4·47-s + 9·49-s + 2·51-s + 6·53-s + 4·57-s − 4·59-s − 2·61-s − 4·63-s + 4·67-s − 8·71-s + 2·73-s − 16·77-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.51·7-s + 1/3·9-s + 1.20·11-s + 0.277·13-s − 0.485·17-s − 0.917·19-s + 0.872·21-s − 0.192·27-s + 1.11·29-s − 0.696·33-s − 1.64·37-s − 0.160·39-s + 0.312·41-s + 0.609·43-s − 0.583·47-s + 9/7·49-s + 0.280·51-s + 0.824·53-s + 0.529·57-s − 0.520·59-s − 0.256·61-s − 0.503·63-s + 0.488·67-s − 0.949·71-s + 0.234·73-s − 1.82·77-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: \(1\)
Selberg data: \((2,\ 31200,\ (\ :1/2),\ -1)\)

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 \( 1 \)
13 \( 1 - T \)
good7 \( 1 + 4 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
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

−15.45746563052728, −14.95555110349994, −14.18365637101948, −13.74156662388855, −13.17251665501460, −12.61370143947619, −12.22218396546880, −11.76022117635415, −11.04586125940362, −10.48979757200102, −10.08716942994901, −9.384660244495443, −8.966798364760700, −8.475853762791593, −7.552272640552805, −6.777888900560858, −6.530357370982660, −6.162577587762142, −5.391000739053855, −4.589075712809548, −3.957566719707330, −3.471921694750173, −2.672572808915003, −1.794142432832619, −0.8626349737607656, 0, 0.8626349737607656, 1.794142432832619, 2.672572808915003, 3.471921694750173, 3.957566719707330, 4.589075712809548, 5.391000739053855, 6.162577587762142, 6.530357370982660, 6.777888900560858, 7.552272640552805, 8.475853762791593, 8.966798364760700, 9.384660244495443, 10.08716942994901, 10.48979757200102, 11.04586125940362, 11.76022117635415, 12.22218396546880, 12.61370143947619, 13.17251665501460, 13.74156662388855, 14.18365637101948, 14.95555110349994, 15.45746563052728

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