Properties

Label 2-15e2-1.1-c11-0-83
Degree $2$
Conductor $225$
Sign $-1$
Analytic cond. $172.877$
Root an. cond. $13.1482$
Motivic weight $11$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 78·2-s + 4.03e3·4-s + 2.77e4·7-s + 1.55e5·8-s − 6.37e5·11-s − 7.66e5·13-s + 2.16e6·14-s + 3.82e6·16-s + 3.08e6·17-s − 1.95e7·19-s − 4.97e7·22-s + 1.53e7·23-s − 5.97e7·26-s + 1.12e8·28-s − 1.07e7·29-s − 5.09e7·31-s − 1.88e7·32-s + 2.40e8·34-s − 6.64e8·37-s − 1.52e9·38-s − 8.98e8·41-s + 9.57e8·43-s − 2.57e9·44-s + 1.19e9·46-s − 1.55e9·47-s − 1.20e9·49-s − 3.09e9·52-s + ⋯
L(s)  = 1  + 1.72·2-s + 1.97·4-s + 0.624·7-s + 1.67·8-s − 1.19·11-s − 0.572·13-s + 1.07·14-s + 0.912·16-s + 0.526·17-s − 1.80·19-s − 2.05·22-s + 0.496·23-s − 0.986·26-s + 1.23·28-s − 0.0973·29-s − 0.319·31-s − 0.0995·32-s + 0.908·34-s − 1.57·37-s − 3.11·38-s − 1.21·41-s + 0.993·43-s − 2.35·44-s + 0.855·46-s − 0.989·47-s − 0.610·49-s − 1.12·52-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $-1$
Analytic conductor: \(172.877\)
Root analytic conductor: \(13.1482\)
Motivic weight: \(11\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 225,\ (\ :11/2),\ -1)\)

Particular Values

\(L(6)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{13}{2})\) 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 \( 1 - 39 p T + p^{11} T^{2} \)
7 \( 1 - 27760 T + p^{11} T^{2} \)
11 \( 1 + 637836 T + p^{11} T^{2} \)
13 \( 1 + 766214 T + p^{11} T^{2} \)
17 \( 1 - 3084354 T + p^{11} T^{2} \)
19 \( 1 + 1026916 p T + p^{11} T^{2} \)
23 \( 1 - 15312360 T + p^{11} T^{2} \)
29 \( 1 + 10751262 T + p^{11} T^{2} \)
31 \( 1 + 50937400 T + p^{11} T^{2} \)
37 \( 1 + 664740830 T + p^{11} T^{2} \)
41 \( 1 + 898833450 T + p^{11} T^{2} \)
43 \( 1 - 957947188 T + p^{11} T^{2} \)
47 \( 1 + 1555741344 T + p^{11} T^{2} \)
53 \( 1 - 3792417030 T + p^{11} T^{2} \)
59 \( 1 + 555306924 T + p^{11} T^{2} \)
61 \( 1 - 4950420998 T + p^{11} T^{2} \)
67 \( 1 + 5292399284 T + p^{11} T^{2} \)
71 \( 1 - 14831086248 T + p^{11} T^{2} \)
73 \( 1 + 13971005210 T + p^{11} T^{2} \)
79 \( 1 - 3720542360 T + p^{11} T^{2} \)
83 \( 1 - 8768454036 T + p^{11} T^{2} \)
89 \( 1 - 25472769174 T + p^{11} T^{2} \)
97 \( 1 - 39092494846 T + p^{11} T^{2} \)
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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

−10.27894798777963299921149730776, −8.587938366514202595616713794262, −7.51101959073324942727161736120, −6.52778554271774839817698478691, −5.36117531591845161734099655617, −4.84529507958256023937915366670, −3.75500849091756485807114740680, −2.64147317186280478211958009929, −1.82142642484890288993755703638, 0, 1.82142642484890288993755703638, 2.64147317186280478211958009929, 3.75500849091756485807114740680, 4.84529507958256023937915366670, 5.36117531591845161734099655617, 6.52778554271774839817698478691, 7.51101959073324942727161736120, 8.587938366514202595616713794262, 10.27894798777963299921149730776

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