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Properties

Label 2-9450-1.1-c1-0-151
Degree $2$
Conductor $9450$
Sign $1$
Analytic cond. $75.4586$
Root an. cond. $8.68669$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $2$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s − 7-s − 8-s − 5·11-s − 5·13-s + 14-s + 16-s − 8·17-s − 8·19-s + 5·22-s − 23-s + 5·26-s − 28-s + 2·29-s − 32-s + 8·34-s + 3·37-s + 8·38-s − 6·41-s − 4·43-s − 5·44-s + 46-s − 9·47-s + 49-s − 5·52-s − 8·53-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.377·7-s − 0.353·8-s − 1.50·11-s − 1.38·13-s + 0.267·14-s + 1/4·16-s − 1.94·17-s − 1.83·19-s + 1.06·22-s − 0.208·23-s + 0.980·26-s − 0.188·28-s + 0.371·29-s − 0.176·32-s + 1.37·34-s + 0.493·37-s + 1.29·38-s − 0.937·41-s − 0.609·43-s − 0.753·44-s + 0.147·46-s − 1.31·47-s + 1/7·49-s − 0.693·52-s − 1.09·53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(9450\)    =    \(2 \cdot 3^{3} \cdot 5^{2} \cdot 7\)
Sign: $1$
Analytic conductor: \(75.4586\)
Root analytic conductor: \(8.68669\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((2,\ 9450,\ (\ :1/2),\ 1)\)

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 \( 1 \)
5 \( 1 \)
7 \( 1 + T \)
good11 \( 1 + 5 T + p T^{2} \)
13 \( 1 + 5 T + p T^{2} \)
17 \( 1 + 8 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 + T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 3 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 9 T + p T^{2} \)
53 \( 1 + 8 T + p T^{2} \)
59 \( 1 + 3 T + p T^{2} \)
61 \( 1 - 5 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 + 13 T + p T^{2} \)
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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

−6.86199187754471504735584741819, −6.68611057070228045468288605906, −5.77119179579693462642658367755, −4.80220830988865414644354478906, −4.43680721525238166857742133686, −3.13467313014539480473105185782, −2.39915626128127124028746469120, −1.94154274456059989670466715676, 0, 0, 1.94154274456059989670466715676, 2.39915626128127124028746469120, 3.13467313014539480473105185782, 4.43680721525238166857742133686, 4.80220830988865414644354478906, 5.77119179579693462642658367755, 6.68611057070228045468288605906, 6.86199187754471504735584741819

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