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Properties

Label 2-169050-1.1-c1-0-114
Degree $2$
Conductor $169050$
Sign $-1$
Analytic cond. $1349.87$
Root an. cond. $36.7405$
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  − 2-s − 3-s + 4-s + 6-s − 8-s + 9-s − 2·11-s − 12-s − 6·13-s + 16-s + 4·17-s − 18-s + 2·19-s + 2·22-s + 23-s + 24-s + 6·26-s − 27-s + 8·29-s − 10·31-s − 32-s + 2·33-s − 4·34-s + 36-s + 2·37-s − 2·38-s + 6·39-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.353·8-s + 1/3·9-s − 0.603·11-s − 0.288·12-s − 1.66·13-s + 1/4·16-s + 0.970·17-s − 0.235·18-s + 0.458·19-s + 0.426·22-s + 0.208·23-s + 0.204·24-s + 1.17·26-s − 0.192·27-s + 1.48·29-s − 1.79·31-s − 0.176·32-s + 0.348·33-s − 0.685·34-s + 1/6·36-s + 0.328·37-s − 0.324·38-s + 0.960·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 169050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(169050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7^{2} \cdot 23\)
Sign: $-1$
Analytic conductor: \(1349.87\)
Root analytic conductor: \(36.7405\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 169050,\ (\ :1/2),\ -1)\)

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 \( 1 \)
23 \( 1 - T \)
good11 \( 1 + 2 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
29 \( 1 - 8 T + p T^{2} \)
31 \( 1 + 10 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 - 10 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 14 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 14 T + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
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

−13.28678541467128, −12.76111178179601, −12.50205670968479, −11.87446758986645, −11.70700030760627, −10.96402430811879, −10.51449119197448, −10.19948414732934, −9.649163604177007, −9.329794007328745, −8.718471185150288, −8.089468690519284, −7.495887701515734, −7.317506112479929, −6.908181133682356, −6.016042233064156, −5.579118618938779, −5.324869844007258, −4.456606456768746, −4.193772310830624, −3.006919692295683, −2.869959016481276, −2.119241418137115, −1.360438673487405, −0.6972886223148885, 0, 0.6972886223148885, 1.360438673487405, 2.119241418137115, 2.869959016481276, 3.006919692295683, 4.193772310830624, 4.456606456768746, 5.324869844007258, 5.579118618938779, 6.016042233064156, 6.908181133682356, 7.317506112479929, 7.495887701515734, 8.089468690519284, 8.718471185150288, 9.329794007328745, 9.649163604177007, 10.19948414732934, 10.51449119197448, 10.96402430811879, 11.70700030760627, 11.87446758986645, 12.50205670968479, 12.76111178179601, 13.28678541467128

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