Properties

Degree $2$
Conductor $1050$
Sign $-1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $1$

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 − 6·13-s − 14-s + 16-s − 2·17-s − 18-s − 4·19-s + 21-s + 4·22-s − 8·23-s − 24-s + 6·26-s + 27-s + 28-s − 2·29-s − 32-s − 4·33-s + 2·34-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 − 1.66·13-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 + 1.17·26-s + 0.192·27-s + 0.188·28-s − 0.371·29-s − 0.176·32-s − 0.696·33-s + 0.342·34-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: $-1$
Motivic weight: \(1\)
Character: $\chi_{1050} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1050,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + 6 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−19.70988803782865, −19.07359182267525, −18.33458532905965, −17.76516030979300, −17.06998648859395, −16.27116890492155, −15.54549458952509, −14.86774343593110, −14.31643658597677, −13.28720768503600, −12.61086972138551, −11.81224716102558, −10.88037870677058, −10.10222911236605, −9.609539393852220, −8.562619095071442, −7.885745778855407, −7.380325109193969, −6.263292807161310, −5.123783494769916, −4.166958040114005, −2.646084597318552, −2.062226348106009, 0, 2.062226348106009, 2.646084597318552, 4.166958040114005, 5.123783494769916, 6.263292807161310, 7.380325109193969, 7.885745778855407, 8.562619095071442, 9.609539393852220, 10.10222911236605, 10.88037870677058, 11.81224716102558, 12.61086972138551, 13.28720768503600, 14.31643658597677, 14.86774343593110, 15.54549458952509, 16.27116890492155, 17.06998648859395, 17.76516030979300, 18.33458532905965, 19.07359182267525, 19.70988803782865

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