Properties

Label 2-1050-1.1-c1-0-11
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 + 6·13-s − 14-s + 16-s + 4·17-s + 18-s − 6·19-s − 21-s + 2·22-s − 8·23-s + 24-s + 6·26-s + 27-s − 28-s + 6·29-s − 2·31-s + 32-s + 2·33-s + 4·34-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 + 1.66·13-s − 0.267·14-s + 1/4·16-s + 0.970·17-s + 0.235·18-s − 1.37·19-s − 0.218·21-s + 0.426·22-s − 1.66·23-s + 0.204·24-s + 1.17·26-s + 0.192·27-s − 0.188·28-s + 1.11·29-s − 0.359·31-s + 0.176·32-s + 0.348·33-s + 0.685·34-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

\(L(1)\) \(\approx\) \(3.184789179\)
\(L(\frac12)\) \(\approx\) \(3.184789179\)
\(L(\frac{3}{2})\) 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 \( 1 - 6 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 + 12 T + p T^{2} \)
83 \( 1 + 8 T + p T^{2} \)
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

−10.04443335843472068951038682726, −8.936983560301573627344332758630, −8.312948613577623088028079646706, −7.37232550640409553455815986114, −6.21997249752887011936324535023, −5.92502214493828437001251745493, −4.28320630078127508245235828197, −3.81978824771939003300116575643, −2.72783326557746750240871274336, −1.42704273090331691804140546291, 1.42704273090331691804140546291, 2.72783326557746750240871274336, 3.81978824771939003300116575643, 4.28320630078127508245235828197, 5.92502214493828437001251745493, 6.21997249752887011936324535023, 7.37232550640409553455815986114, 8.312948613577623088028079646706, 8.936983560301573627344332758630, 10.04443335843472068951038682726

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