Properties

Label 2-9225-1.1-c1-0-15
Degree $2$
Conductor $9225$
Sign $1$
Analytic cond. $73.6619$
Root an. cond. $8.58265$
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  + 0.470·2-s − 1.77·4-s − 3.30·7-s − 1.77·8-s + 1.47·11-s + 0.249·13-s − 1.55·14-s + 2.71·16-s − 5.02·17-s − 2.24·19-s + 0.692·22-s − 6.24·23-s + 0.117·26-s + 5.88·28-s + 2.41·29-s − 2.89·31-s + 4.83·32-s − 2.36·34-s − 9.71·37-s − 1.05·38-s − 41-s + 10.8·43-s − 2.61·44-s − 2.94·46-s − 4.08·47-s + 3.94·49-s − 0.443·52-s + ⋯
L(s)  = 1  + 0.332·2-s − 0.889·4-s − 1.25·7-s − 0.628·8-s + 0.443·11-s + 0.0690·13-s − 0.416·14-s + 0.679·16-s − 1.21·17-s − 0.515·19-s + 0.147·22-s − 1.30·23-s + 0.0229·26-s + 1.11·28-s + 0.447·29-s − 0.520·31-s + 0.855·32-s − 0.405·34-s − 1.59·37-s − 0.171·38-s − 0.156·41-s + 1.65·43-s − 0.394·44-s − 0.433·46-s − 0.596·47-s + 0.563·49-s − 0.0614·52-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(9225\)    =    \(3^{2} \cdot 5^{2} \cdot 41\)
Sign: $1$
Analytic conductor: \(73.6619\)
Root analytic conductor: \(8.58265\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 9225,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5957499882\)
\(L(\frac12)\) \(\approx\) \(0.5957499882\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
41 \( 1 + T \)
good2 \( 1 - 0.470T + 2T^{2} \)
7 \( 1 + 3.30T + 7T^{2} \)
11 \( 1 - 1.47T + 11T^{2} \)
13 \( 1 - 0.249T + 13T^{2} \)
17 \( 1 + 5.02T + 17T^{2} \)
19 \( 1 + 2.24T + 19T^{2} \)
23 \( 1 + 6.24T + 23T^{2} \)
29 \( 1 - 2.41T + 29T^{2} \)
31 \( 1 + 2.89T + 31T^{2} \)
37 \( 1 + 9.71T + 37T^{2} \)
43 \( 1 - 10.8T + 43T^{2} \)
47 \( 1 + 4.08T + 47T^{2} \)
53 \( 1 + 1.43T + 53T^{2} \)
59 \( 1 + 2.61T + 59T^{2} \)
61 \( 1 + 5.71T + 61T^{2} \)
67 \( 1 + 15.9T + 67T^{2} \)
71 \( 1 - 10.5T + 71T^{2} \)
73 \( 1 + 9.39T + 73T^{2} \)
79 \( 1 + 0.560T + 79T^{2} \)
83 \( 1 - 3.80T + 83T^{2} \)
89 \( 1 + 4.11T + 89T^{2} \)
97 \( 1 - 7.19T + 97T^{2} \)
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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

−7.71867517840511940915717622149, −6.85200912228170181819051321231, −6.23751442509504906033064402605, −5.80402678113419651883437728305, −4.79906737085838717398538508047, −4.17267591046663824033441567129, −3.60033425364963245897447180437, −2.83688917604169698129118004199, −1.78126318132749241025653759188, −0.34199062774096657850812815169, 0.34199062774096657850812815169, 1.78126318132749241025653759188, 2.83688917604169698129118004199, 3.60033425364963245897447180437, 4.17267591046663824033441567129, 4.79906737085838717398538508047, 5.80402678113419651883437728305, 6.23751442509504906033064402605, 6.85200912228170181819051321231, 7.71867517840511940915717622149

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