Properties

Label 2-15e2-1.1-c3-0-1
Degree $2$
Conductor $225$
Sign $1$
Analytic cond. $13.2754$
Root an. cond. $3.64354$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4.70·2-s + 14.1·4-s − 16.2·7-s − 28.7·8-s + 40.2·11-s + 19.7·13-s + 76.2·14-s + 22.1·16-s − 83.0·17-s − 48.8·19-s − 189.·22-s − 1.61·23-s − 93.0·26-s − 228.·28-s + 24.5·29-s − 12.4·31-s + 125.·32-s + 390.·34-s + 325.·37-s + 229.·38-s + 242.·41-s + 367.·43-s + 567.·44-s + 7.58·46-s + 204.·47-s − 80.2·49-s + 279.·52-s + ⋯
L(s)  = 1  − 1.66·2-s + 1.76·4-s − 0.875·7-s − 1.26·8-s + 1.10·11-s + 0.422·13-s + 1.45·14-s + 0.345·16-s − 1.18·17-s − 0.589·19-s − 1.83·22-s − 0.0146·23-s − 0.701·26-s − 1.54·28-s + 0.157·29-s − 0.0719·31-s + 0.694·32-s + 1.96·34-s + 1.44·37-s + 0.980·38-s + 0.923·41-s + 1.30·43-s + 1.94·44-s + 0.0242·46-s + 0.634·47-s − 0.233·49-s + 0.744·52-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(13.2754\)
Root analytic conductor: \(3.64354\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(0.6600603069\)
\(L(\frac12)\) \(\approx\) \(0.6600603069\)
\(L(\frac{5}{2})\) 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 \( 1 \)
good2 \( 1 + 4.70T + 8T^{2} \)
7 \( 1 + 16.2T + 343T^{2} \)
11 \( 1 - 40.2T + 1.33e3T^{2} \)
13 \( 1 - 19.7T + 2.19e3T^{2} \)
17 \( 1 + 83.0T + 4.91e3T^{2} \)
19 \( 1 + 48.8T + 6.85e3T^{2} \)
23 \( 1 + 1.61T + 1.21e4T^{2} \)
29 \( 1 - 24.5T + 2.43e4T^{2} \)
31 \( 1 + 12.4T + 2.97e4T^{2} \)
37 \( 1 - 325.T + 5.06e4T^{2} \)
41 \( 1 - 242.T + 6.89e4T^{2} \)
43 \( 1 - 367.T + 7.95e4T^{2} \)
47 \( 1 - 204.T + 1.03e5T^{2} \)
53 \( 1 - 61.5T + 1.48e5T^{2} \)
59 \( 1 - 112.T + 2.05e5T^{2} \)
61 \( 1 - 477.T + 2.26e5T^{2} \)
67 \( 1 - 558.T + 3.00e5T^{2} \)
71 \( 1 + 558.T + 3.57e5T^{2} \)
73 \( 1 - 1.01e3T + 3.89e5T^{2} \)
79 \( 1 - 1.15e3T + 4.93e5T^{2} \)
83 \( 1 - 1.15e3T + 5.71e5T^{2} \)
89 \( 1 + 96.9T + 7.04e5T^{2} \)
97 \( 1 + 1.15e3T + 9.12e5T^{2} \)
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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

−11.35228670765698464519941519497, −10.69245416093424217944452775945, −9.507293083474311064757666705894, −9.103740165166302557944028315442, −8.069119485440187005198235074778, −6.84146947712091373497428141875, −6.23560279506738679801959823004, −4.07381497356330039798219886972, −2.34631098794546676435365039157, −0.76763121822473264237611057094, 0.76763121822473264237611057094, 2.34631098794546676435365039157, 4.07381497356330039798219886972, 6.23560279506738679801959823004, 6.84146947712091373497428141875, 8.069119485440187005198235074778, 9.103740165166302557944028315442, 9.507293083474311064757666705894, 10.69245416093424217944452775945, 11.35228670765698464519941519497

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