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Properties

Label 2-169050-1.1-c1-0-70
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 3-s + 4-s + 6-s + 8-s + 9-s + 12-s − 4·13-s + 16-s + 18-s − 2·19-s − 23-s + 24-s − 4·26-s + 27-s + 6·29-s − 2·31-s + 32-s + 36-s + 10·37-s − 2·38-s − 4·39-s − 6·41-s + 4·43-s − 46-s + 6·47-s + 48-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 + 0.288·12-s − 1.10·13-s + 1/4·16-s + 0.235·18-s − 0.458·19-s − 0.208·23-s + 0.204·24-s − 0.784·26-s + 0.192·27-s + 1.11·29-s − 0.359·31-s + 0.176·32-s + 1/6·36-s + 1.64·37-s − 0.324·38-s − 0.640·39-s − 0.937·41-s + 0.609·43-s − 0.147·46-s + 0.875·47-s + 0.144·48-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: \(0\)
Selberg data: \((2,\ 169050,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(5.103674232\)
\(L(\frac12)\) \(\approx\) \(5.103674232\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 - 8 T + p T^{2} \)
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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

−13.30519454044430, −12.77594241654326, −12.23505192233718, −12.10270022865794, −11.39828454436008, −10.88146120615214, −10.35027052241135, −9.991532256589131, −9.309882152845506, −9.076062565095513, −8.189117563787593, −7.953713078134887, −7.435116547639356, −6.772639077058095, −6.506188688236026, −5.816431957226966, −5.183877933745804, −4.799975366543878, −4.160441946530364, −3.800574613440426, −3.053858997783579, −2.430087574340452, −2.260014259032830, −1.314875777848118, −0.5508556152887117, 0.5508556152887117, 1.314875777848118, 2.260014259032830, 2.430087574340452, 3.053858997783579, 3.800574613440426, 4.160441946530364, 4.799975366543878, 5.183877933745804, 5.816431957226966, 6.506188688236026, 6.772639077058095, 7.435116547639356, 7.953713078134887, 8.189117563787593, 9.076062565095513, 9.309882152845506, 9.991532256589131, 10.35027052241135, 10.88146120615214, 11.39828454436008, 12.10270022865794, 12.23505192233718, 12.77594241654326, 13.30519454044430

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