Properties

Degree 4
Conductor $ 5^{2} \cdot 17^{2} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 2

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s − 4·3-s + 4-s − 2·5-s + 8·6-s − 4·7-s + 8·9-s + 4·10-s − 8·11-s − 4·12-s + 8·14-s + 8·15-s + 16-s − 2·17-s − 16·18-s − 2·20-s + 16·21-s + 16·22-s − 4·23-s + 3·25-s − 12·27-s − 4·28-s − 4·29-s − 16·30-s + 2·32-s + 32·33-s + 4·34-s + ⋯
L(s)  = 1  − 1.41·2-s − 2.30·3-s + 1/2·4-s − 0.894·5-s + 3.26·6-s − 1.51·7-s + 8/3·9-s + 1.26·10-s − 2.41·11-s − 1.15·12-s + 2.13·14-s + 2.06·15-s + 1/4·16-s − 0.485·17-s − 3.77·18-s − 0.447·20-s + 3.49·21-s + 3.41·22-s − 0.834·23-s + 3/5·25-s − 2.30·27-s − 0.755·28-s − 0.742·29-s − 2.92·30-s + 0.353·32-s + 5.57·33-s + 0.685·34-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 7225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 7225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(7225\)    =    \(5^{2} \cdot 17^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{7225} (1, \cdot )$
Sato-Tate  :  $G_{3,3}$
primitive  :  no
self-dual  :  yes
analytic rank  =  2
Selberg data  =  $(4,\ 7225,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{5,\;17\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{5,\;17\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad5$C_1$ \( ( 1 + T )^{2} \)
17$C_1$ \( ( 1 + T )^{2} \)
good2$D_{4}$ \( 1 + p T + 3 T^{2} + p^{2} T^{3} + p^{2} T^{4} \)
3$C_2^2$ \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
7$D_{4}$ \( 1 + 4 T + 16 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 + 8 T + 36 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
13$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \)
19$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \)
23$D_{4}$ \( 1 + 4 T + 48 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
31$C_2^2$ \( 1 + 44 T^{2} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + 4 T + 6 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 - 4 T + 14 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 4 T + 58 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 4 T + 90 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - 12 T + 110 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 24 T + 254 T^{2} + 24 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 - 4 T + 94 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 12 T + 162 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
71$C_2^2$ \( 1 + 124 T^{2} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 4 T + 142 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 8 T + 172 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 4 T + 42 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 16 T + 210 T^{2} + 16 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 4 T + 166 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−17.6772889811, −16.8916357753, −16.7617476276, −16.044020326, −15.9522969109, −15.5151197908, −14.9502151208, −13.5820805748, −13.2618240913, −12.5780527616, −12.3177553969, −11.7201521045, −11.0630836973, −10.5784756992, −10.391268185, −9.65365847432, −9.20371420294, −8.22335296348, −7.75778713541, −7.10272384564, −6.31406013724, −5.77708222231, −5.19016061385, −4.27247643896, −2.95634545112, 0, 0, 2.95634545112, 4.27247643896, 5.19016061385, 5.77708222231, 6.31406013724, 7.10272384564, 7.75778713541, 8.22335296348, 9.20371420294, 9.65365847432, 10.391268185, 10.5784756992, 11.0630836973, 11.7201521045, 12.3177553969, 12.5780527616, 13.2618240913, 13.5820805748, 14.9502151208, 15.5151197908, 15.9522969109, 16.044020326, 16.7617476276, 16.8916357753, 17.6772889811

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