Properties

Degree 2
Conductor $ 5^{2} \cdot 7 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

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  =  0
Selberg data  =  $(2,\ 175,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.347442534$
$L(\frac12)$  $\approx$  $2.347442534$
$L(\frac{3}{2})$   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

−19.11583525256306, −18.34331539125220, −16.99770359975083, −16.08511794812033, −15.03712770989199, −14.56543966402684, −13.57311735160890, −13.01254965419347, −12.04804768145327, −11.05181111943764, −9.734749354117360, −8.569343753087953, −7.384517030180517, −5.938987404347633, −5.149361617353738, −3.577525821264665, −2.764724220886489, 2.764724220886489, 3.577525821264665, 5.149361617353738, 5.938987404347633, 7.384517030180517, 8.569343753087953, 9.734749354117360, 11.05181111943764, 12.04804768145327, 13.01254965419347, 13.57311735160890, 14.56543966402684, 15.03712770989199, 16.08511794812033, 16.99770359975083, 18.34331539125220, 19.11583525256306

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