Properties

Label 2-9450-1.1-c1-0-111
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s + 7-s − 8-s − 2·11-s + 4.16·13-s − 14-s + 16-s − 7.32·17-s + 3·19-s + 2·22-s − 1.16·23-s − 4.16·26-s + 28-s − 1.83·29-s − 6.32·31-s − 32-s + 7.32·34-s + 7.48·37-s − 3·38-s − 4·41-s + 3.16·43-s − 2·44-s + 1.16·46-s + 10.4·47-s + 49-s + 4.16·52-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.5·4-s + 0.377·7-s − 0.353·8-s − 0.603·11-s + 1.15·13-s − 0.267·14-s + 0.250·16-s − 1.77·17-s + 0.688·19-s + 0.426·22-s − 0.242·23-s − 0.816·26-s + 0.188·28-s − 0.341·29-s − 1.13·31-s − 0.176·32-s + 1.25·34-s + 1.23·37-s − 0.486·38-s − 0.624·41-s + 0.482·43-s − 0.301·44-s + 0.171·46-s + 1.52·47-s + 0.142·49-s + 0.577·52-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: \(1\)
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 \( 1 + 2T + 11T^{2} \)
13 \( 1 - 4.16T + 13T^{2} \)
17 \( 1 + 7.32T + 17T^{2} \)
19 \( 1 - 3T + 19T^{2} \)
23 \( 1 + 1.16T + 23T^{2} \)
29 \( 1 + 1.83T + 29T^{2} \)
31 \( 1 + 6.32T + 31T^{2} \)
37 \( 1 - 7.48T + 37T^{2} \)
41 \( 1 + 4T + 41T^{2} \)
43 \( 1 - 3.16T + 43T^{2} \)
47 \( 1 - 10.4T + 47T^{2} \)
53 \( 1 - 0.162T + 53T^{2} \)
59 \( 1 - 0.324T + 59T^{2} \)
61 \( 1 + 3.83T + 61T^{2} \)
67 \( 1 + 3.48T + 67T^{2} \)
71 \( 1 - 10.3T + 71T^{2} \)
73 \( 1 + 4.32T + 73T^{2} \)
79 \( 1 + 14.4T + 79T^{2} \)
83 \( 1 + 3.16T + 83T^{2} \)
89 \( 1 + 5T + 89T^{2} \)
97 \( 1 + 11.4T + 97T^{2} \)
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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

−7.40192135823526484139726646766, −6.86462405734080933227230339599, −5.99521021703325236608977722028, −5.51332439630514213567480174561, −4.49566717085629860074630671956, −3.84286672613746588369990259052, −2.82332828624089217476612858898, −2.07422030889149788830524973187, −1.18477920583352514228386994337, 0, 1.18477920583352514228386994337, 2.07422030889149788830524973187, 2.82332828624089217476612858898, 3.84286672613746588369990259052, 4.49566717085629860074630671956, 5.51332439630514213567480174561, 5.99521021703325236608977722028, 6.86462405734080933227230339599, 7.40192135823526484139726646766

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