Properties

Label 2-1050-1.1-c3-0-38
Degree $2$
Conductor $1050$
Sign $-1$
Analytic cond. $61.9520$
Root an. cond. $7.87095$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s − 3·3-s + 4·4-s + 6·6-s + 7·7-s − 8·8-s + 9·9-s − 43·11-s − 12·12-s − 18·13-s − 14·14-s + 16·16-s + 92·17-s − 18·18-s + 6·19-s − 21·21-s + 86·22-s + 129·23-s + 24·24-s + 36·26-s − 27·27-s + 28·28-s − 111·29-s − 142·31-s − 32·32-s + 129·33-s − 184·34-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s − 1.17·11-s − 0.288·12-s − 0.384·13-s − 0.267·14-s + 1/4·16-s + 1.31·17-s − 0.235·18-s + 0.0724·19-s − 0.218·21-s + 0.833·22-s + 1.16·23-s + 0.204·24-s + 0.271·26-s − 0.192·27-s + 0.188·28-s − 0.710·29-s − 0.822·31-s − 0.176·32-s + 0.680·33-s − 0.928·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-1$
Analytic conductor: \(61.9520\)
Root analytic conductor: \(7.87095\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1050,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + p T \)
3 \( 1 + p T \)
5 \( 1 \)
7 \( 1 - p T \)
good11 \( 1 + 43 T + p^{3} T^{2} \)
13 \( 1 + 18 T + p^{3} T^{2} \)
17 \( 1 - 92 T + p^{3} T^{2} \)
19 \( 1 - 6 T + p^{3} T^{2} \)
23 \( 1 - 129 T + p^{3} T^{2} \)
29 \( 1 + 111 T + p^{3} T^{2} \)
31 \( 1 + 142 T + p^{3} T^{2} \)
37 \( 1 + 173 T + p^{3} T^{2} \)
41 \( 1 + 128 T + p^{3} T^{2} \)
43 \( 1 + 217 T + p^{3} T^{2} \)
47 \( 1 - 194 T + p^{3} T^{2} \)
53 \( 1 - 622 T + p^{3} T^{2} \)
59 \( 1 + 22 T + p^{3} T^{2} \)
61 \( 1 - 160 T + p^{3} T^{2} \)
67 \( 1 - 189 T + p^{3} T^{2} \)
71 \( 1 - 769 T + p^{3} T^{2} \)
73 \( 1 - 652 T + p^{3} T^{2} \)
79 \( 1 + 773 T + p^{3} T^{2} \)
83 \( 1 - 314 T + p^{3} T^{2} \)
89 \( 1 - 470 T + p^{3} T^{2} \)
97 \( 1 + 470 T + p^{3} T^{2} \)
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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

−9.191298340022373193847209639141, −8.205060854074687948929982923181, −7.50429112786008367856229922508, −6.82339224654713639224193060204, −5.45734209012122063932193769003, −5.19070344217998001249121221753, −3.62772798462865466465745822339, −2.43909531652078488933273999817, −1.19713050641540895559396634533, 0, 1.19713050641540895559396634533, 2.43909531652078488933273999817, 3.62772798462865466465745822339, 5.19070344217998001249121221753, 5.45734209012122063932193769003, 6.82339224654713639224193060204, 7.50429112786008367856229922508, 8.205060854074687948929982923181, 9.191298340022373193847209639141

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