Properties

Degree 2
Conductor $ 5^{2} \cdot 7 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s − 3-s + 2·4-s + 2·6-s + 7-s − 2·9-s − 3·11-s − 2·12-s − 13-s − 2·14-s − 4·16-s − 7·17-s + 4·18-s − 21-s + 6·22-s − 6·23-s + 2·26-s + 5·27-s + 2·28-s − 5·29-s + 2·31-s + 8·32-s + 3·33-s + 14·34-s − 4·36-s − 2·37-s + 39-s + ⋯
L(s)  = 1  − 1.41·2-s − 0.577·3-s + 4-s + 0.816·6-s + 0.377·7-s − 2/3·9-s − 0.904·11-s − 0.577·12-s − 0.277·13-s − 0.534·14-s − 16-s − 1.69·17-s + 0.942·18-s − 0.218·21-s + 1.27·22-s − 1.25·23-s + 0.392·26-s + 0.962·27-s + 0.377·28-s − 0.928·29-s + 0.359·31-s + 1.41·32-s + 0.522·33-s + 2.40·34-s − 2/3·36-s − 0.328·37-s + 0.160·39-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(175\)    =    \(5^{2} \cdot 7\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{175} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 175,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{5,\;7\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{5,\;7\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad5 \( 1 \)
7 \( 1 - T \)
good2 \( 1 + p T + p T^{2} \)
3 \( 1 + T + p T^{2} \)
11 \( 1 + 3 T + p T^{2} \)
13 \( 1 + T + p T^{2} \)
17 \( 1 + 7 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 5 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 3 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 10 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + 5 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 + 7 T + p T^{2} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−19.28072942207594, −18.23202788870947, −17.75806690057465, −17.14356288790802, −16.19433536467873, −15.47094180098410, −14.14373622549030, −13.09935125315941, −11.74927591551528, −10.99407496803787, −10.28604192239362, −9.098174271283234, −8.274076028912949, −7.303300206181466, −5.998486980227912, −4.631713104794750, −2.252674004710408, 0, 2.252674004710408, 4.631713104794750, 5.998486980227912, 7.303300206181466, 8.274076028912949, 9.098174271283234, 10.28604192239362, 10.99407496803787, 11.74927591551528, 13.09935125315941, 14.14373622549030, 15.47094180098410, 16.19433536467873, 17.14356288790802, 17.75806690057465, 18.23202788870947, 19.28072942207594

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