Properties

Label 2-1050-1.1-c1-0-3
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

\(L(1)\) \(\approx\) \(1.339863646\)
\(L(\frac12)\) \(\approx\) \(1.339863646\)
\(L(\frac{3}{2})\) 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

−9.838791532098356959217399335575, −9.314276201894361189726268747264, −8.132788049862507970445322945220, −7.54392364458419480987676267402, −6.97587151947823251519146924106, −5.59816972376066845528951357350, −4.79565646870238517074069242438, −3.14773582507890044917606243068, −2.66687668466195404295404364048, −0.989278977507431980256878998913, 0.989278977507431980256878998913, 2.66687668466195404295404364048, 3.14773582507890044917606243068, 4.79565646870238517074069242438, 5.59816972376066845528951357350, 6.97587151947823251519146924106, 7.54392364458419480987676267402, 8.132788049862507970445322945220, 9.314276201894361189726268747264, 9.838791532098356959217399335575

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