Properties

Label 2-1050-1.1-c1-0-12
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 + 6·17-s + 18-s + 21-s − 4·22-s + 8·23-s + 24-s + 2·26-s + 27-s + 28-s + 10·29-s − 8·31-s + 32-s − 4·33-s + 6·34-s + 36-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 + 1.45·17-s + 0.235·18-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 + 1.85·29-s − 1.43·31-s + 0.176·32-s − 0.696·33-s + 1.02·34-s + 1/6·36-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

\(L(1)\) \(\approx\) \(3.206165521\)
\(L(\frac12)\) \(\approx\) \(3.206165521\)
\(L(\frac{3}{2})\) 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 \( 1 - 6 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 - 10 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−10.11558774468085427099984179567, −8.970038638688739077275471188963, −8.117271450797248149047450674807, −7.49183328013247917757345935069, −6.51937316169267569375873124058, −5.35085092662211501216455986720, −4.81962159294781503660641716359, −3.46413801666508801779103816982, −2.83397636642881070566824261870, −1.42737253017477970910017504740, 1.42737253017477970910017504740, 2.83397636642881070566824261870, 3.46413801666508801779103816982, 4.81962159294781503660641716359, 5.35085092662211501216455986720, 6.51937316169267569375873124058, 7.49183328013247917757345935069, 8.117271450797248149047450674807, 8.970038638688739077275471188963, 10.11558774468085427099984179567

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