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Properties

Label 2-300-3.2-c0-0-1
Degree $2$
Conductor $300$
Sign $1$
Analytic cond. $0.149719$
Root an. cond. $0.386936$
Motivic weight $0$
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 − 7-s + 9-s − 13-s − 19-s − 21-s + 27-s − 31-s + 2·37-s − 39-s − 43-s − 57-s − 61-s − 63-s − 67-s + 2·73-s + 2·79-s + 81-s + 91-s − 93-s − 97-s + 2·103-s − 109-s + 2·111-s − 117-s + ⋯
L(s)  = 1  + 3-s − 7-s + 9-s − 13-s − 19-s − 21-s + 27-s − 31-s + 2·37-s − 39-s − 43-s − 57-s − 61-s − 63-s − 67-s + 2·73-s + 2·79-s + 81-s + 91-s − 93-s − 97-s + 2·103-s − 109-s + 2·111-s − 117-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(0.149719\)
Root analytic conductor: \(0.386936\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{300} (101, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 300,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9379306138\)
\(L(\frac12)\) \(\approx\) \(0.9379306138\)
\(L(1)\) 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 \( 1 \)
good7 \( 1 + T + T^{2} \)
11 \( ( 1 - T )( 1 + T ) \)
13 \( 1 + T + T^{2} \)
17 \( ( 1 - T )( 1 + T ) \)
19 \( 1 + T + T^{2} \)
23 \( ( 1 - T )( 1 + T ) \)
29 \( ( 1 - T )( 1 + T ) \)
31 \( 1 + T + T^{2} \)
37 \( ( 1 - T )^{2} \)
41 \( ( 1 - T )( 1 + T ) \)
43 \( 1 + T + T^{2} \)
47 \( ( 1 - T )( 1 + T ) \)
53 \( ( 1 - T )( 1 + T ) \)
59 \( ( 1 - T )( 1 + T ) \)
61 \( 1 + T + T^{2} \)
67 \( 1 + T + T^{2} \)
71 \( ( 1 - T )( 1 + T ) \)
73 \( ( 1 - T )^{2} \)
79 \( ( 1 - T )^{2} \)
83 \( ( 1 - T )( 1 + T ) \)
89 \( ( 1 - T )( 1 + T ) \)
97 \( 1 + T + T^{2} \)
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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

−12.28635711682333976456896360080, −10.86036084061802502300255538697, −9.813505761984259359602911118440, −9.299156206700353354343999261248, −8.179375125771904294432326961111, −7.22578870015588797157442167915, −6.25493360265909365346641648673, −4.65000897702590843147352066318, −3.44829529350115176093652141218, −2.30049063152907025395205531779, 2.30049063152907025395205531779, 3.44829529350115176093652141218, 4.65000897702590843147352066318, 6.25493360265909365346641648673, 7.22578870015588797157442167915, 8.179375125771904294432326961111, 9.299156206700353354343999261248, 9.813505761984259359602911118440, 10.86036084061802502300255538697, 12.28635711682333976456896360080

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