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Properties

Label 2-303450-1.1-c1-0-134
Degree $2$
Conductor $303450$
Sign $-1$
Analytic cond. $2423.06$
Root an. cond. $49.2245$
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 − 7-s − 8-s + 9-s + 3·11-s − 12-s + 14-s + 16-s − 18-s + 21-s − 3·22-s + 4·23-s + 24-s − 27-s − 28-s + 3·29-s + 5·31-s − 32-s − 3·33-s + 36-s + 4·37-s − 42-s − 6·43-s + 3·44-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 + 0.904·11-s − 0.288·12-s + 0.267·14-s + 1/4·16-s − 0.235·18-s + 0.218·21-s − 0.639·22-s + 0.834·23-s + 0.204·24-s − 0.192·27-s − 0.188·28-s + 0.557·29-s + 0.898·31-s − 0.176·32-s − 0.522·33-s + 1/6·36-s + 0.657·37-s − 0.154·42-s − 0.914·43-s + 0.452·44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 303450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 303450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(303450\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7 \cdot 17^{2}\)
Sign: $-1$
Analytic conductor: \(2423.06\)
Root analytic conductor: \(49.2245\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 303450,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
17 \( 1 \)
good11 \( 1 - 3 T + p T^{2} \)
13 \( 1 + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 - 5 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 6 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 - 7 T + p T^{2} \)
59 \( 1 - 7 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 + 6 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 3 T + p T^{2} \)
79 \( 1 - 5 T + p T^{2} \)
83 \( 1 - 8 T + p T^{2} \)
89 \( 1 + 16 T + p T^{2} \)
97 \( 1 + 7 T + p T^{2} \)
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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

−12.90337335287741, −12.21034660667604, −11.96738208118487, −11.45587428861119, −11.12390139969581, −10.62459372072234, −10.04148530697042, −9.704446968792022, −9.418164232294542, −8.637210759768047, −8.457417285395852, −7.847541284722488, −7.237516231121828, −6.712678277865739, −6.516038486347201, −6.072068388837486, −5.271312286835857, −5.010914576938852, −4.240995251189096, −3.781067552699453, −3.141729287981186, −2.602494659009291, −1.914217179657744, −1.185843583587011, −0.8227542818965131, 0, 0.8227542818965131, 1.185843583587011, 1.914217179657744, 2.602494659009291, 3.141729287981186, 3.781067552699453, 4.240995251189096, 5.010914576938852, 5.271312286835857, 6.072068388837486, 6.516038486347201, 6.712678277865739, 7.237516231121828, 7.847541284722488, 8.457417285395852, 8.637210759768047, 9.418164232294542, 9.704446968792022, 10.04148530697042, 10.62459372072234, 11.12390139969581, 11.45587428861119, 11.96738208118487, 12.21034660667604, 12.90337335287741

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