Properties

Label 2-1050-1.1-c1-0-4
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 + 4·11-s + 12-s + 2·13-s + 14-s + 16-s − 2·17-s − 18-s − 4·19-s − 21-s − 4·22-s + 8·23-s − 24-s − 2·26-s + 27-s − 28-s + 6·29-s − 8·31-s − 32-s + 4·33-s + 2·34-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 + 1.20·11-s + 0.288·12-s + 0.554·13-s + 0.267·14-s + 1/4·16-s − 0.485·17-s − 0.235·18-s − 0.917·19-s − 0.218·21-s − 0.852·22-s + 1.66·23-s − 0.204·24-s − 0.392·26-s + 0.192·27-s − 0.188·28-s + 1.11·29-s − 1.43·31-s − 0.176·32-s + 0.696·33-s + 0.342·34-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

\(L(1)\) \(\approx\) \(1.500296459\)
\(L(\frac12)\) \(\approx\) \(1.500296459\)
\(L(\frac{3}{2})\) 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 \( 1 - 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
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

−9.625809353569461036085862289118, −8.993802635293947884746939912145, −8.601351251693854628821228011897, −7.41750004971397288593425742052, −6.73892644153891083442310873080, −5.95251363997533671735290275834, −4.45776004306556863633827269854, −3.51293692634747222502503411630, −2.39285800255617234808224415611, −1.08656149587708795900736124088, 1.08656149587708795900736124088, 2.39285800255617234808224415611, 3.51293692634747222502503411630, 4.45776004306556863633827269854, 5.95251363997533671735290275834, 6.73892644153891083442310873080, 7.41750004971397288593425742052, 8.601351251693854628821228011897, 8.993802635293947884746939912145, 9.625809353569461036085862289118

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