Properties

Label 2-7650-1.1-c1-0-60
Degree $2$
Conductor $7650$
Sign $1$
Analytic cond. $61.0855$
Root an. cond. $7.81572$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s + 8-s + 6·11-s + 3·13-s + 16-s − 17-s − 7·19-s + 6·22-s + 8·23-s + 3·26-s + 5·29-s + 5·31-s + 32-s − 34-s + 8·37-s − 7·38-s − 4·43-s + 6·44-s + 8·46-s − 3·47-s − 7·49-s + 3·52-s − 9·53-s + 5·58-s − 5·59-s − 3·61-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.353·8-s + 1.80·11-s + 0.832·13-s + 1/4·16-s − 0.242·17-s − 1.60·19-s + 1.27·22-s + 1.66·23-s + 0.588·26-s + 0.928·29-s + 0.898·31-s + 0.176·32-s − 0.171·34-s + 1.31·37-s − 1.13·38-s − 0.609·43-s + 0.904·44-s + 1.17·46-s − 0.437·47-s − 49-s + 0.416·52-s − 1.23·53-s + 0.656·58-s − 0.650·59-s − 0.384·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(7650\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 17\)
Sign: $1$
Analytic conductor: \(61.0855\)
Root analytic conductor: \(7.81572\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 7650,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.103661850\)
\(L(\frac12)\) \(\approx\) \(4.103661850\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 \)
5 \( 1 \)
17 \( 1 + T \)
good7 \( 1 + p T^{2} \)
11 \( 1 - 6 T + p T^{2} \)
13 \( 1 - 3 T + p T^{2} \)
19 \( 1 + 7 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 - 5 T + p T^{2} \)
31 \( 1 - 5 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 3 T + p T^{2} \)
53 \( 1 + 9 T + p T^{2} \)
59 \( 1 + 5 T + p T^{2} \)
61 \( 1 + 3 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 - 15 T + p T^{2} \)
73 \( 1 + 11 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - T + p T^{2} \)
97 \( 1 + 9 T + p T^{2} \)
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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

−7.86625989498839697894186665041, −6.71978821832837603727796473612, −6.52921660631650771229096311130, −6.00423531650877286662157691066, −4.75275142198392183299279007503, −4.44319372834745597801581895719, −3.59130951406283393164894958848, −2.90022234396185536904625056998, −1.78274366372577389926427627202, −0.979933555115995156564765897609, 0.979933555115995156564765897609, 1.78274366372577389926427627202, 2.90022234396185536904625056998, 3.59130951406283393164894958848, 4.44319372834745597801581895719, 4.75275142198392183299279007503, 6.00423531650877286662157691066, 6.52921660631650771229096311130, 6.71978821832837603727796473612, 7.86625989498839697894186665041

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