Properties

Label 2-15e2-1.1-c7-0-24
Degree $2$
Conductor $225$
Sign $-1$
Analytic cond. $70.2866$
Root an. cond. $8.38371$
Motivic weight $7$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 13·2-s + 41·4-s − 1.38e3·7-s + 1.13e3·8-s + 3.30e3·11-s − 8.50e3·13-s + 1.79e4·14-s − 1.99e4·16-s − 9.99e3·17-s + 4.12e4·19-s − 4.29e4·22-s + 8.41e4·23-s + 1.10e5·26-s − 5.65e4·28-s − 1.32e5·29-s − 5.58e4·31-s + 1.14e5·32-s + 1.29e5·34-s − 2.28e5·37-s − 5.36e5·38-s + 1.39e5·41-s + 7.55e5·43-s + 1.35e5·44-s − 1.09e6·46-s + 8.36e5·47-s + 1.08e6·49-s − 3.48e5·52-s + ⋯
L(s)  = 1  − 1.14·2-s + 0.320·4-s − 1.52·7-s + 0.780·8-s + 0.748·11-s − 1.07·13-s + 1.74·14-s − 1.21·16-s − 0.493·17-s + 1.37·19-s − 0.860·22-s + 1.44·23-s + 1.23·26-s − 0.487·28-s − 1.01·29-s − 0.336·31-s + 0.618·32-s + 0.566·34-s − 0.740·37-s − 1.58·38-s + 0.316·41-s + 1.44·43-s + 0.239·44-s − 1.65·46-s + 1.17·47-s + 1.31·49-s − 0.343·52-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $-1$
Analytic conductor: \(70.2866\)
Root analytic conductor: \(8.38371\)
Motivic weight: \(7\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 225,\ (\ :7/2),\ -1)\)

Particular Values

\(L(4)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{9}{2})\) 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 \( 1 + 13 T + p^{7} T^{2} \)
7 \( 1 + 1380 T + p^{7} T^{2} \)
11 \( 1 - 3304 T + p^{7} T^{2} \)
13 \( 1 + 8506 T + p^{7} T^{2} \)
17 \( 1 + 9994 T + p^{7} T^{2} \)
19 \( 1 - 41236 T + p^{7} T^{2} \)
23 \( 1 - 84120 T + p^{7} T^{2} \)
29 \( 1 + 132802 T + p^{7} T^{2} \)
31 \( 1 + 1800 p T + p^{7} T^{2} \)
37 \( 1 + 228170 T + p^{7} T^{2} \)
41 \( 1 - 139670 T + p^{7} T^{2} \)
43 \( 1 - 755492 T + p^{7} T^{2} \)
47 \( 1 - 836984 T + p^{7} T^{2} \)
53 \( 1 - 1641650 T + p^{7} T^{2} \)
59 \( 1 - 989656 T + p^{7} T^{2} \)
61 \( 1 + 1658162 T + p^{7} T^{2} \)
67 \( 1 - 4523844 T + p^{7} T^{2} \)
71 \( 1 - 389408 T + p^{7} T^{2} \)
73 \( 1 + 5617330 T + p^{7} T^{2} \)
79 \( 1 - 3901080 T + p^{7} T^{2} \)
83 \( 1 + 9394116 T + p^{7} T^{2} \)
89 \( 1 + 2803746 T + p^{7} T^{2} \)
97 \( 1 + 5099426 T + p^{7} T^{2} \)
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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

−10.13590289248672239655328947755, −9.361714635681398398542040571914, −8.985565275117669913857607564969, −7.38157383507033487949220520736, −6.91926637290393902454941417450, −5.43090453293092857091980034841, −3.94768014861018863201046212804, −2.63023448921915820619949355358, −1.03394155082932235885602762885, 0, 1.03394155082932235885602762885, 2.63023448921915820619949355358, 3.94768014861018863201046212804, 5.43090453293092857091980034841, 6.91926637290393902454941417450, 7.38157383507033487949220520736, 8.985565275117669913857607564969, 9.361714635681398398542040571914, 10.13590289248672239655328947755

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