Properties

Label 2-1050-1.1-c1-0-9
Degree $2$
Conductor $1050$
Sign $1$
Analytic cond. $8.38429$
Root an. cond. $2.89556$
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 − 3-s + 4-s − 6-s + 7-s + 8-s + 9-s − 2·11-s − 12-s + 7·13-s + 14-s + 16-s − 7·17-s + 18-s + 8·19-s − 21-s − 2·22-s − 5·23-s − 24-s + 7·26-s − 27-s + 28-s + 9·29-s + 31-s + 32-s + 2·33-s − 7·34-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 − 0.603·11-s − 0.288·12-s + 1.94·13-s + 0.267·14-s + 1/4·16-s − 1.69·17-s + 0.235·18-s + 1.83·19-s − 0.218·21-s − 0.426·22-s − 1.04·23-s − 0.204·24-s + 1.37·26-s − 0.192·27-s + 0.188·28-s + 1.67·29-s + 0.179·31-s + 0.176·32-s + 0.348·33-s − 1.20·34-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $1$
Analytic conductor: \(8.38429\)
Root analytic conductor: \(2.89556\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1050} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1050,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.276979907\)
\(L(\frac12)\) \(\approx\) \(2.276979907\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
5 \( 1 \)
7 \( 1 - T \)
good11 \( 1 + 2 T + p T^{2} \)
13 \( 1 - 7 T + p T^{2} \)
17 \( 1 + 7 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 + 5 T + p T^{2} \)
29 \( 1 - 9 T + p T^{2} \)
31 \( 1 - T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 11 T + p T^{2} \)
43 \( 1 - 3 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 - 3 T + p T^{2} \)
59 \( 1 - 7 T + p T^{2} \)
61 \( 1 + 5 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + 4 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 6 T + p T^{2} \)
83 \( 1 + 9 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−10.23154085150512267010470695300, −9.023211319798265371761026653318, −8.176838270825692892122658475126, −7.23729673110132585489087841457, −6.26006454492861389761955890441, −5.69311262378796882208602612047, −4.66900070901730711067447771935, −3.88497779591437622791413052836, −2.62503810000977451640909828816, −1.18268320323446649342437002114, 1.18268320323446649342437002114, 2.62503810000977451640909828816, 3.88497779591437622791413052836, 4.66900070901730711067447771935, 5.69311262378796882208602612047, 6.26006454492861389761955890441, 7.23729673110132585489087841457, 8.176838270825692892122658475126, 9.023211319798265371761026653318, 10.23154085150512267010470695300

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