Properties

Label 2-1200-3.2-c0-0-0
Degree $2$
Conductor $1200$
Sign $1$
Analytic cond. $0.598878$
Root an. cond. $0.773872$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1200\)    =    \(2^{4} \cdot 3 \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(0.598878\)
Root analytic conductor: \(0.773872\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1200} (401, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1200,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8670312834\)
\(L(\frac12)\) \(\approx\) \(0.8670312834\)
\(L(1)\) 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

−9.950126173511207971518112912092, −9.391846209416087629579946162636, −8.052275601766956634635657358492, −7.52314732748779362199215745432, −6.58747390409545588061962370111, −5.60654943959490241212002341952, −4.90021730363715398245167051424, −4.17904794327763203216460541414, −2.57929737615652012801569709618, −1.18698551234494244795327854849, 1.18698551234494244795327854849, 2.57929737615652012801569709618, 4.17904794327763203216460541414, 4.90021730363715398245167051424, 5.60654943959490241212002341952, 6.58747390409545588061962370111, 7.52314732748779362199215745432, 8.052275601766956634635657358492, 9.391846209416087629579946162636, 9.950126173511207971518112912092

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