Properties

Label 2-2175-1.1-c1-0-76
Degree $2$
Conductor $2175$
Sign $-1$
Analytic cond. $17.3674$
Root an. cond. $4.16742$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 2·4-s + 2·7-s + 9-s + 11-s − 2·12-s − 6·13-s + 4·16-s − 4·17-s − 2·19-s + 2·21-s − 3·23-s + 27-s − 4·28-s + 29-s − 4·31-s + 33-s − 2·36-s + 3·37-s − 6·39-s + 7·41-s − 5·43-s − 2·44-s − 6·47-s + 4·48-s − 3·49-s − 4·51-s + ⋯
L(s)  = 1  + 0.577·3-s − 4-s + 0.755·7-s + 1/3·9-s + 0.301·11-s − 0.577·12-s − 1.66·13-s + 16-s − 0.970·17-s − 0.458·19-s + 0.436·21-s − 0.625·23-s + 0.192·27-s − 0.755·28-s + 0.185·29-s − 0.718·31-s + 0.174·33-s − 1/3·36-s + 0.493·37-s − 0.960·39-s + 1.09·41-s − 0.762·43-s − 0.301·44-s − 0.875·47-s + 0.577·48-s − 3/7·49-s − 0.560·51-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2175\)    =    \(3 \cdot 5^{2} \cdot 29\)
Sign: $-1$
Analytic conductor: \(17.3674\)
Root analytic conductor: \(4.16742\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 2175,\ (\ :1/2),\ -1)\)

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)$
bad3 \( 1 - T \)
5 \( 1 \)
29 \( 1 - T \)
good2 \( 1 + p T^{2} \)
7 \( 1 - 2 T + p T^{2} \)
11 \( 1 - T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 + 3 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 3 T + p T^{2} \)
41 \( 1 - 7 T + p T^{2} \)
43 \( 1 + 5 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 + 13 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 - 10 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 + 3 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 9 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 + 17 T + p T^{2} \)
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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

−8.616145362225899285091988408871, −8.061638128858403786474137913687, −7.36278754958494017460252627790, −6.37285416545749887240215303884, −5.17522095389622481613177994570, −4.61302926044132737393909951738, −3.92515695251871556359483309630, −2.69254658465815446366551411536, −1.68273659030778080859428751300, 0, 1.68273659030778080859428751300, 2.69254658465815446366551411536, 3.92515695251871556359483309630, 4.61302926044132737393909951738, 5.17522095389622481613177994570, 6.37285416545749887240215303884, 7.36278754958494017460252627790, 8.061638128858403786474137913687, 8.616145362225899285091988408871

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