Properties

Label 2-169050-1.1-c1-0-46
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 + 6·13-s + 16-s + 4·17-s − 18-s − 4·19-s − 23-s + 24-s − 6·26-s − 27-s − 8·29-s + 4·31-s − 32-s − 4·34-s + 36-s − 2·37-s + 4·38-s − 6·39-s − 4·43-s + 46-s − 4·47-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 + 1.66·13-s + 1/4·16-s + 0.970·17-s − 0.235·18-s − 0.917·19-s − 0.208·23-s + 0.204·24-s − 1.17·26-s − 0.192·27-s − 1.48·29-s + 0.718·31-s − 0.176·32-s − 0.685·34-s + 1/6·36-s − 0.328·37-s + 0.648·38-s − 0.960·39-s − 0.609·43-s + 0.147·46-s − 0.583·47-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: $\chi_{169050} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 169050,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.410698130\)
\(L(\frac12)\) \(\approx\) \(1.410698130\)
\(L(\frac{3}{2})\) 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 \( 1 - 6 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 - 12 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 6 T + p T^{2} \)
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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

−13.21218216676593, −12.72557001606588, −12.13958101733741, −11.70167638950286, −11.20993820017257, −10.91726454358759, −10.31433145604775, −10.05105819285942, −9.377330236247143, −8.938389108735882, −8.383659002104756, −7.994212874983716, −7.542540996932709, −6.743120855492745, −6.490025885272996, −5.964170464388128, −5.455019544485715, −4.973837597246290, −4.056418313384104, −3.715897427538085, −3.170762453315190, −2.251442436588974, −1.704375425275522, −1.089124213474610, −0.4532476709774553, 0.4532476709774553, 1.089124213474610, 1.704375425275522, 2.251442436588974, 3.170762453315190, 3.715897427538085, 4.056418313384104, 4.973837597246290, 5.455019544485715, 5.964170464388128, 6.490025885272996, 6.743120855492745, 7.542540996932709, 7.994212874983716, 8.383659002104756, 8.938389108735882, 9.377330236247143, 10.05105819285942, 10.31433145604775, 10.91726454358759, 11.20993820017257, 11.70167638950286, 12.13958101733741, 12.72557001606588, 13.21218216676593

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