Properties

Label 2-9450-1.1-c1-0-47
Degree $2$
Conductor $9450$
Sign $1$
Analytic cond. $75.4586$
Root an. cond. $8.68669$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s + 7-s + 8-s − 1.73·11-s − 5.46·13-s + 14-s + 16-s + 4·17-s + 5.73·19-s − 1.73·22-s + 2.46·23-s − 5.46·26-s + 28-s − 1.46·29-s − 0.267·31-s + 32-s + 4·34-s + 3.19·37-s + 5.73·38-s + 3.92·41-s + 4.53·43-s − 1.73·44-s + 2.46·46-s + 11.4·47-s + 49-s − 5.46·52-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s + 0.377·7-s + 0.353·8-s − 0.522·11-s − 1.51·13-s + 0.267·14-s + 0.250·16-s + 0.970·17-s + 1.31·19-s − 0.369·22-s + 0.513·23-s − 1.07·26-s + 0.188·28-s − 0.271·29-s − 0.0481·31-s + 0.176·32-s + 0.685·34-s + 0.525·37-s + 0.929·38-s + 0.613·41-s + 0.691·43-s − 0.261·44-s + 0.363·46-s + 1.67·47-s + 0.142·49-s − 0.757·52-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: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 9450,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.413056448\)
\(L(\frac12)\) \(\approx\) \(3.413056448\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.73T + 11T^{2} \)
13 \( 1 + 5.46T + 13T^{2} \)
17 \( 1 - 4T + 17T^{2} \)
19 \( 1 - 5.73T + 19T^{2} \)
23 \( 1 - 2.46T + 23T^{2} \)
29 \( 1 + 1.46T + 29T^{2} \)
31 \( 1 + 0.267T + 31T^{2} \)
37 \( 1 - 3.19T + 37T^{2} \)
41 \( 1 - 3.92T + 41T^{2} \)
43 \( 1 - 4.53T + 43T^{2} \)
47 \( 1 - 11.4T + 47T^{2} \)
53 \( 1 + 13.8T + 53T^{2} \)
59 \( 1 + 4.92T + 59T^{2} \)
61 \( 1 + 13.8T + 61T^{2} \)
67 \( 1 - 12.3T + 67T^{2} \)
71 \( 1 + 4.26T + 71T^{2} \)
73 \( 1 + 16.9T + 73T^{2} \)
79 \( 1 - 8.92T + 79T^{2} \)
83 \( 1 - 9.46T + 83T^{2} \)
89 \( 1 + T + 89T^{2} \)
97 \( 1 - 6.92T + 97T^{2} \)
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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

−7.48547119668401162695854659721, −7.28206586646266552736610757147, −6.11424666223687944322368273447, −5.54626575236342589924684132457, −4.91194540177050100715608673949, −4.42252992477271427311391874327, −3.30074612997481487973740371713, −2.80928999505812828281901754044, −1.90922888309098944470021687333, −0.794369016615572623477208408104, 0.794369016615572623477208408104, 1.90922888309098944470021687333, 2.80928999505812828281901754044, 3.30074612997481487973740371713, 4.42252992477271427311391874327, 4.91194540177050100715608673949, 5.54626575236342589924684132457, 6.11424666223687944322368273447, 7.28206586646266552736610757147, 7.48547119668401162695854659721

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