Properties

Label 2-300-3.2-c2-0-7
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 − 2·7-s + 9·9-s + 22·13-s + 26·19-s − 6·21-s + 27·27-s − 46·31-s − 26·37-s + 66·39-s + 22·43-s − 45·49-s + 78·57-s + 74·61-s − 18·63-s − 122·67-s + 46·73-s − 142·79-s + 81·81-s − 44·91-s − 138·93-s − 2·97-s − 194·103-s − 214·109-s − 78·111-s + 198·117-s + ⋯
L(s)  = 1  + 3-s − 2/7·7-s + 9-s + 1.69·13-s + 1.36·19-s − 2/7·21-s + 27-s − 1.48·31-s − 0.702·37-s + 1.69·39-s + 0.511·43-s − 0.918·49-s + 1.36·57-s + 1.21·61-s − 2/7·63-s − 1.82·67-s + 0.630·73-s − 1.79·79-s + 81-s − 0.483·91-s − 1.48·93-s − 0.0206·97-s − 1.88·103-s − 1.96·109-s − 0.702·111-s + 1.69·117-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

\(L(\frac{3}{2})\) \(\approx\) \(2.401540729\)
\(L(\frac12)\) \(\approx\) \(2.401540729\)
\(L(2)\) 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 \( 1 + 2 T + p^{2} T^{2} \)
11 \( ( 1 - p T )( 1 + p T ) \)
13 \( 1 - 22 T + p^{2} T^{2} \)
17 \( ( 1 - p T )( 1 + p T ) \)
19 \( 1 - 26 T + p^{2} T^{2} \)
23 \( ( 1 - p T )( 1 + p T ) \)
29 \( ( 1 - p T )( 1 + p T ) \)
31 \( 1 + 46 T + p^{2} T^{2} \)
37 \( 1 + 26 T + p^{2} T^{2} \)
41 \( ( 1 - p T )( 1 + p T ) \)
43 \( 1 - 22 T + p^{2} T^{2} \)
47 \( ( 1 - p T )( 1 + p T ) \)
53 \( ( 1 - p T )( 1 + p T ) \)
59 \( ( 1 - p T )( 1 + p T ) \)
61 \( 1 - 74 T + p^{2} T^{2} \)
67 \( 1 + 122 T + p^{2} T^{2} \)
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 \( 1 + 2 T + p^{2} T^{2} \)
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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

−11.46656385207626903932464774790, −10.49110043613348289599393355082, −9.449991295604027734411408633800, −8.746294124187591753618348514257, −7.78266523924152754912803783596, −6.79057395138763157228416778818, −5.53056390782487474059852767197, −3.96331443121751516890018649961, −3.10959065995811925214115925516, −1.45396119538811166520156230439, 1.45396119538811166520156230439, 3.10959065995811925214115925516, 3.96331443121751516890018649961, 5.53056390782487474059852767197, 6.79057395138763157228416778818, 7.78266523924152754912803783596, 8.746294124187591753618348514257, 9.449991295604027734411408633800, 10.49110043613348289599393355082, 11.46656385207626903932464774790

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