Properties

Label 2-21175-1.1-c1-0-29
Degree $2$
Conductor $21175$
Sign $-1$
Analytic cond. $169.083$
Root an. cond. $13.0032$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 2·4-s + 7-s − 2·9-s + 2·12-s + 5·13-s + 4·16-s + 3·17-s − 2·19-s − 21-s + 6·23-s + 5·27-s − 2·28-s − 3·29-s − 4·31-s + 4·36-s − 2·37-s − 5·39-s + 12·41-s − 10·43-s − 9·47-s − 4·48-s + 49-s − 3·51-s − 10·52-s − 12·53-s + 2·57-s + ⋯
L(s)  = 1  − 0.577·3-s − 4-s + 0.377·7-s − 2/3·9-s + 0.577·12-s + 1.38·13-s + 16-s + 0.727·17-s − 0.458·19-s − 0.218·21-s + 1.25·23-s + 0.962·27-s − 0.377·28-s − 0.557·29-s − 0.718·31-s + 2/3·36-s − 0.328·37-s − 0.800·39-s + 1.87·41-s − 1.52·43-s − 1.31·47-s − 0.577·48-s + 1/7·49-s − 0.420·51-s − 1.38·52-s − 1.64·53-s + 0.264·57-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(21175\)    =    \(5^{2} \cdot 7 \cdot 11^{2}\)
Sign: $-1$
Analytic conductor: \(169.083\)
Root analytic conductor: \(13.0032\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{21175} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 21175,\ (\ :1/2),\ -1)\)

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)$
bad5 \( 1 \)
7 \( 1 - T \)
11 \( 1 \)
good2 \( 1 + p T^{2} \)
3 \( 1 + T + p T^{2} \)
13 \( 1 - 5 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 12 T + p T^{2} \)
43 \( 1 + 10 T + p T^{2} \)
47 \( 1 + 9 T + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 12 T + p T^{2} \)
97 \( 1 - T + p T^{2} \)
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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

−15.98533003011212, −15.17394815236956, −14.68481433131774, −14.25184244223825, −13.69434943594772, −13.08343763980005, −12.69469682558460, −12.08641891727972, −11.20966224974127, −11.09125644096789, −10.49071041227718, −9.654844342171572, −9.141699579621685, −8.609613521313064, −8.116110338414978, −7.545287541523227, −6.534251942296786, −6.090183137666915, −5.362475568262401, −5.034029190498415, −4.250254156426593, −3.495707698006425, −3.007791731866615, −1.670040502129005, −0.9673420945380721, 0, 0.9673420945380721, 1.670040502129005, 3.007791731866615, 3.495707698006425, 4.250254156426593, 5.034029190498415, 5.362475568262401, 6.090183137666915, 6.534251942296786, 7.545287541523227, 8.116110338414978, 8.609613521313064, 9.141699579621685, 9.654844342171572, 10.49071041227718, 11.09125644096789, 11.20966224974127, 12.08641891727972, 12.69469682558460, 13.08343763980005, 13.69434943594772, 14.25184244223825, 14.68481433131774, 15.17394815236956, 15.98533003011212

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