Properties

Label 2-300-3.2-c2-0-8
Degree $2$
Conductor $300$
Sign $1$
Analytic cond. $8.17440$
Root an. cond. $2.85909$
Motivic weight $2$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·3-s + 13·7-s + 9·9-s − 23·13-s + 11·19-s + 39·21-s + 27·27-s + 59·31-s − 26·37-s − 69·39-s − 83·43-s + 120·49-s + 33·57-s − 121·61-s + 117·63-s + 13·67-s + 46·73-s − 142·79-s + 81·81-s − 299·91-s + 177·93-s − 167·97-s − 194·103-s + 71·109-s − 78·111-s − 207·117-s + ⋯
L(s)  = 1  + 3-s + 13/7·7-s + 9-s − 1.76·13-s + 0.578·19-s + 13/7·21-s + 27-s + 1.90·31-s − 0.702·37-s − 1.76·39-s − 1.93·43-s + 2.44·49-s + 0.578·57-s − 1.98·61-s + 13/7·63-s + 0.194·67-s + 0.630·73-s − 1.79·79-s + 81-s − 3.28·91-s + 1.90·93-s − 1.72·97-s − 1.88·103-s + 0.651·109-s − 0.702·111-s − 1.76·117-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(8.17440\)
Root analytic conductor: \(2.85909\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: $\chi_{300} (101, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 300,\ (\ :1),\ 1)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.624844647\)
\(L(\frac12)\) \(\approx\) \(2.624844647\)
\(L(2)\) 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 \( 1 - p T \)
5 \( 1 \)
good7 \( 1 - 13 T + p^{2} T^{2} \)
11 \( ( 1 - p T )( 1 + p T ) \)
13 \( 1 + 23 T + p^{2} T^{2} \)
17 \( ( 1 - p T )( 1 + p T ) \)
19 \( 1 - 11 T + p^{2} T^{2} \)
23 \( ( 1 - p T )( 1 + p T ) \)
29 \( ( 1 - p T )( 1 + p T ) \)
31 \( 1 - 59 T + p^{2} T^{2} \)
37 \( 1 + 26 T + p^{2} T^{2} \)
41 \( ( 1 - p T )( 1 + p T ) \)
43 \( 1 + 83 T + p^{2} T^{2} \)
47 \( ( 1 - p T )( 1 + p T ) \)
53 \( ( 1 - p T )( 1 + p T ) \)
59 \( ( 1 - p T )( 1 + p T ) \)
61 \( 1 + 121 T + p^{2} T^{2} \)
67 \( 1 - 13 T + p^{2} T^{2} \)
71 \( ( 1 - p T )( 1 + p T ) \)
73 \( 1 - 46 T + p^{2} T^{2} \)
79 \( 1 + 142 T + p^{2} T^{2} \)
83 \( ( 1 - p T )( 1 + p T ) \)
89 \( ( 1 - p T )( 1 + p T ) \)
97 \( 1 + 167 T + p^{2} T^{2} \)
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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

−11.67435859645297621403052964776, −10.43408980830569027442693636095, −9.619731914503858134084662166870, −8.460112652955810907226618264267, −7.84967749173925088632960273151, −7.01369605587529416804517469514, −5.11185734724104878503839759333, −4.45522778289589384118515576210, −2.77684077723795404072676856845, −1.60138498571055517302612665151, 1.60138498571055517302612665151, 2.77684077723795404072676856845, 4.45522778289589384118515576210, 5.11185734724104878503839759333, 7.01369605587529416804517469514, 7.84967749173925088632960273151, 8.460112652955810907226618264267, 9.619731914503858134084662166870, 10.43408980830569027442693636095, 11.67435859645297621403052964776

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