Properties

Label 2-15e2-1.1-c3-0-8
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  + 3.16·2-s + 2.00·4-s + 15·7-s − 18.9·8-s + 63.2·11-s + 35·13-s + 47.4·14-s − 76·16-s + 88.5·17-s + 91·19-s + 200·22-s − 113.·23-s + 110.·26-s + 30.0·28-s − 63.2·29-s − 147·31-s − 88.5·32-s + 280·34-s + 370·37-s + 287.·38-s − 442.·41-s + 335·43-s + 126.·44-s − 360·46-s − 177.·47-s − 118·49-s + 70.0·52-s + ⋯
L(s)  = 1  + 1.11·2-s + 0.250·4-s + 0.809·7-s − 0.838·8-s + 1.73·11-s + 0.746·13-s + 0.905·14-s − 1.18·16-s + 1.26·17-s + 1.09·19-s + 1.93·22-s − 1.03·23-s + 0.834·26-s + 0.202·28-s − 0.404·29-s − 0.851·31-s − 0.489·32-s + 1.41·34-s + 1.64·37-s + 1.22·38-s − 1.68·41-s + 1.18·43-s + 0.433·44-s − 1.15·46-s − 0.549·47-s − 0.344·49-s + 0.186·52-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

\(L(2)\) \(\approx\) \(3.366982914\)
\(L(\frac12)\) \(\approx\) \(3.366982914\)
\(L(\frac{5}{2})\) 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 \( 1 - 3.16T + 8T^{2} \)
7 \( 1 - 15T + 343T^{2} \)
11 \( 1 - 63.2T + 1.33e3T^{2} \)
13 \( 1 - 35T + 2.19e3T^{2} \)
17 \( 1 - 88.5T + 4.91e3T^{2} \)
19 \( 1 - 91T + 6.85e3T^{2} \)
23 \( 1 + 113.T + 1.21e4T^{2} \)
29 \( 1 + 63.2T + 2.43e4T^{2} \)
31 \( 1 + 147T + 2.97e4T^{2} \)
37 \( 1 - 370T + 5.06e4T^{2} \)
41 \( 1 + 442.T + 6.89e4T^{2} \)
43 \( 1 - 335T + 7.95e4T^{2} \)
47 \( 1 + 177.T + 1.03e5T^{2} \)
53 \( 1 - 88.5T + 1.48e5T^{2} \)
59 \( 1 + 885.T + 2.05e5T^{2} \)
61 \( 1 - 427T + 2.26e5T^{2} \)
67 \( 1 - 15T + 3.00e5T^{2} \)
71 \( 1 + 63.2T + 3.57e5T^{2} \)
73 \( 1 + 70T + 3.89e5T^{2} \)
79 \( 1 + 876T + 4.93e5T^{2} \)
83 \( 1 + 531.T + 5.71e5T^{2} \)
89 \( 1 + 7.04e5T^{2} \)
97 \( 1 + 1.08e3T + 9.12e5T^{2} \)
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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

−11.79500268335814591671195664500, −11.40357111009650033936710953608, −9.789727895342154361174443507442, −8.890664913287295193998741414151, −7.70180819667229364920721371508, −6.31317906131062753753098744230, −5.46386451791316632437277941259, −4.23444943724310348445779606696, −3.39606622370191707662506827238, −1.38666394890942631136185453236, 1.38666394890942631136185453236, 3.39606622370191707662506827238, 4.23444943724310348445779606696, 5.46386451791316632437277941259, 6.31317906131062753753098744230, 7.70180819667229364920721371508, 8.890664913287295193998741414151, 9.789727895342154361174443507442, 11.40357111009650033936710953608, 11.79500268335814591671195664500

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