Properties

Degree $2$
Conductor $175$
Sign $-1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 2·4-s − 7-s − 2·9-s − 3·11-s + 2·12-s − 5·13-s + 4·16-s − 3·17-s + 2·19-s + 21-s + 6·23-s + 5·27-s + 2·28-s + 3·29-s − 4·31-s + 3·33-s + 4·36-s − 2·37-s + 5·39-s − 12·41-s + 10·43-s + 6·44-s − 9·47-s − 4·48-s + 49-s + 3·51-s + ⋯
L(s)  = 1  − 0.577·3-s − 4-s − 0.377·7-s − 2/3·9-s − 0.904·11-s + 0.577·12-s − 1.38·13-s + 16-s − 0.727·17-s + 0.458·19-s + 0.218·21-s + 1.25·23-s + 0.962·27-s + 0.377·28-s + 0.557·29-s − 0.718·31-s + 0.522·33-s + 2/3·36-s − 0.328·37-s + 0.800·39-s − 1.87·41-s + 1.52·43-s + 0.904·44-s − 1.31·47-s − 0.577·48-s + 1/7·49-s + 0.420·51-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

Degree: \(2\)
Conductor: \(175\)    =    \(5^{2} \cdot 7\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{175} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 175,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
good2 \( 1 + p T^{2} \)
3 \( 1 + T + p T^{2} \)
11 \( 1 + 3 T + p T^{2} \)
13 \( 1 + 5 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 12 T + p T^{2} \)
43 \( 1 - 10 T + p T^{2} \)
47 \( 1 + 9 T + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 12 T + p T^{2} \)
97 \( 1 - T + p T^{2} \)
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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

−19.29975422298916, −18.47561052890729, −17.51730526638904, −17.13633307361341, −16.04905149867832, −14.93587348540493, −14.07833352570527, −13.06641347252683, −12.38708666154083, −11.22233622493929, −10.17729908145777, −9.261964485898248, −8.214592059880155, −6.934540116371023, −5.450201785974024, −4.786860739351209, −2.987040885382746, 0, 2.987040885382746, 4.786860739351209, 5.450201785974024, 6.934540116371023, 8.214592059880155, 9.261964485898248, 10.17729908145777, 11.22233622493929, 12.38708666154083, 13.06641347252683, 14.07833352570527, 14.93587348540493, 16.04905149867832, 17.13633307361341, 17.51730526638904, 18.47561052890729, 19.29975422298916

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