Properties

Label 2-31200-1.1-c1-0-26
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 + 6·11-s + 13-s + 2·17-s + 6·19-s + 2·21-s + 27-s − 6·29-s − 6·31-s + 6·33-s − 2·37-s + 39-s − 10·41-s + 8·43-s + 6·47-s − 3·49-s + 2·51-s − 6·53-s + 6·57-s + 6·59-s − 10·61-s + 2·63-s + 2·67-s + 14·71-s + 14·73-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.755·7-s + 1/3·9-s + 1.80·11-s + 0.277·13-s + 0.485·17-s + 1.37·19-s + 0.436·21-s + 0.192·27-s − 1.11·29-s − 1.07·31-s + 1.04·33-s − 0.328·37-s + 0.160·39-s − 1.56·41-s + 1.21·43-s + 0.875·47-s − 3/7·49-s + 0.280·51-s − 0.824·53-s + 0.794·57-s + 0.781·59-s − 1.28·61-s + 0.251·63-s + 0.244·67-s + 1.66·71-s + 1.63·73-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

\(L(1)\) \(\approx\) \(4.453173023\)
\(L(\frac12)\) \(\approx\) \(4.453173023\)
\(L(\frac{3}{2})\) 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 \( 1 - 6 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 6 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 - 14 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−14.95841146586852, −14.47230871073813, −14.10526654882210, −13.74512852814452, −13.04916770296524, −12.24336800255845, −12.04478204464354, −11.23147116433374, −11.07207721830812, −10.15408806099677, −9.486046899860879, −9.184840211391039, −8.708387523424683, −7.867433301011438, −7.584736208307097, −6.863423668008145, −6.332608206096642, −5.473183370147315, −5.057301468721336, −4.124910150770040, −3.662001749368041, −3.191439546919219, −2.033336556207387, −1.546347174208768, −0.8391456568847315, 0.8391456568847315, 1.546347174208768, 2.033336556207387, 3.191439546919219, 3.662001749368041, 4.124910150770040, 5.057301468721336, 5.473183370147315, 6.332608206096642, 6.863423668008145, 7.584736208307097, 7.867433301011438, 8.708387523424683, 9.184840211391039, 9.486046899860879, 10.15408806099677, 11.07207721830812, 11.23147116433374, 12.04478204464354, 12.24336800255845, 13.04916770296524, 13.74512852814452, 14.10526654882210, 14.47230871073813, 14.95841146586852

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