Properties

Label 2-31200-1.1-c1-0-31
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 + 13-s − 3·19-s − 4·21-s + 4·23-s + 27-s − 29-s − 8·31-s + 37-s + 39-s + 41-s − 6·43-s + 11·47-s + 9·49-s + 3·53-s − 3·57-s + 10·59-s + 4·61-s − 4·63-s − 13·67-s + 4·69-s + 9·71-s − 3·79-s + 81-s − 2·83-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.51·7-s + 1/3·9-s + 0.277·13-s − 0.688·19-s − 0.872·21-s + 0.834·23-s + 0.192·27-s − 0.185·29-s − 1.43·31-s + 0.164·37-s + 0.160·39-s + 0.156·41-s − 0.914·43-s + 1.60·47-s + 9/7·49-s + 0.412·53-s − 0.397·57-s + 1.30·59-s + 0.512·61-s − 0.503·63-s − 1.58·67-s + 0.481·69-s + 1.06·71-s − 0.337·79-s + 1/9·81-s − 0.219·83-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 \( 1 + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 3 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - T + p T^{2} \)
41 \( 1 - T + p T^{2} \)
43 \( 1 + 6 T + p T^{2} \)
47 \( 1 - 11 T + p T^{2} \)
53 \( 1 - 3 T + p T^{2} \)
59 \( 1 - 10 T + p T^{2} \)
61 \( 1 - 4 T + p T^{2} \)
67 \( 1 + 13 T + p T^{2} \)
71 \( 1 - 9 T + p T^{2} \)
73 \( 1 + p T^{2} \)
79 \( 1 + 3 T + p T^{2} \)
83 \( 1 + 2 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 - 8 T + p T^{2} \)
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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

−15.41443855701677, −14.72698899220208, −14.47383557443083, −13.53588958723562, −13.25538113848622, −12.87504555534292, −12.33278957242244, −11.72458411111291, −10.92314633301433, −10.49778025675717, −9.931663567339949, −9.302284622457139, −8.983870362548026, −8.437807869100230, −7.641760148682328, −7.028258363888981, −6.681019757956649, −5.934615299336914, −5.425711832304923, −4.514580035720354, −3.767302117413220, −3.421058723609902, −2.660716029643702, −2.048569261014872, −0.9758961575919157, 0, 0.9758961575919157, 2.048569261014872, 2.660716029643702, 3.421058723609902, 3.767302117413220, 4.514580035720354, 5.425711832304923, 5.934615299336914, 6.681019757956649, 7.028258363888981, 7.641760148682328, 8.437807869100230, 8.983870362548026, 9.302284622457139, 9.931663567339949, 10.49778025675717, 10.92314633301433, 11.72458411111291, 12.33278957242244, 12.87504555534292, 13.25538113848622, 13.53588958723562, 14.47383557443083, 14.72698899220208, 15.41443855701677

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