Properties

Degree $2$
Conductor $1225$
Sign $-1$
Motivic weight $3$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·2-s + 2·3-s + 8·4-s + 8·6-s − 23·9-s + 32·11-s + 16·12-s − 38·13-s − 64·16-s + 26·17-s − 92·18-s − 100·19-s + 128·22-s + 78·23-s − 152·26-s − 100·27-s − 50·29-s + 108·31-s − 256·32-s + 64·33-s + 104·34-s − 184·36-s − 266·37-s − 400·38-s − 76·39-s − 22·41-s − 442·43-s + ⋯
L(s)  = 1  + 1.41·2-s + 0.384·3-s + 4-s + 0.544·6-s − 0.851·9-s + 0.877·11-s + 0.384·12-s − 0.810·13-s − 16-s + 0.370·17-s − 1.20·18-s − 1.20·19-s + 1.24·22-s + 0.707·23-s − 1.14·26-s − 0.712·27-s − 0.320·29-s + 0.625·31-s − 1.41·32-s + 0.337·33-s + 0.524·34-s − 0.851·36-s − 1.18·37-s − 1.70·38-s − 0.312·39-s − 0.0838·41-s − 1.56·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1225\)    =    \(5^{2} \cdot 7^{2}\)
Sign: $-1$
Motivic weight: \(3\)
Character: $\chi_{1225} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1225,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
good2 \( 1 - p^{2} T + p^{3} T^{2} \)
3 \( 1 - 2 T + p^{3} T^{2} \)
11 \( 1 - 32 T + p^{3} T^{2} \)
13 \( 1 + 38 T + p^{3} T^{2} \)
17 \( 1 - 26 T + p^{3} T^{2} \)
19 \( 1 + 100 T + p^{3} T^{2} \)
23 \( 1 - 78 T + p^{3} T^{2} \)
29 \( 1 + 50 T + p^{3} T^{2} \)
31 \( 1 - 108 T + p^{3} T^{2} \)
37 \( 1 + 266 T + p^{3} T^{2} \)
41 \( 1 + 22 T + p^{3} T^{2} \)
43 \( 1 + 442 T + p^{3} T^{2} \)
47 \( 1 + 514 T + p^{3} T^{2} \)
53 \( 1 + 2 T + p^{3} T^{2} \)
59 \( 1 + 500 T + p^{3} T^{2} \)
61 \( 1 - 518 T + p^{3} T^{2} \)
67 \( 1 + 126 T + p^{3} T^{2} \)
71 \( 1 - 412 T + p^{3} T^{2} \)
73 \( 1 + 878 T + p^{3} T^{2} \)
79 \( 1 - 600 T + p^{3} T^{2} \)
83 \( 1 - 282 T + p^{3} T^{2} \)
89 \( 1 - 150 T + p^{3} T^{2} \)
97 \( 1 - 386 T + p^{3} T^{2} \)
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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

−8.887742131654996575089293281878, −8.162477297389007211227102478631, −6.91234646997756984659847251517, −6.33411754980782401853834991051, −5.34850217767977901736317400574, −4.64835741008777700973313574605, −3.64268285220326236839184857359, −2.94748180853785410463070920938, −1.89544522072685515185572394616, 0, 1.89544522072685515185572394616, 2.94748180853785410463070920938, 3.64268285220326236839184857359, 4.64835741008777700973313574605, 5.34850217767977901736317400574, 6.33411754980782401853834991051, 6.91234646997756984659847251517, 8.162477297389007211227102478631, 8.887742131654996575089293281878

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