Properties

Label 2-4275-1.1-c1-0-99
Degree $2$
Conductor $4275$
Sign $-1$
Analytic cond. $34.1360$
Root an. cond. $5.84260$
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  − 2·4-s + 7-s − 3·11-s + 4·13-s + 4·16-s − 3·17-s + 19-s − 2·28-s − 6·29-s − 4·31-s − 2·37-s + 6·41-s + 43-s + 6·44-s − 3·47-s − 6·49-s − 8·52-s + 12·53-s + 6·59-s − 61-s − 8·64-s + 4·67-s + 6·68-s − 6·71-s + 7·73-s − 2·76-s − 3·77-s + ⋯
L(s)  = 1  − 4-s + 0.377·7-s − 0.904·11-s + 1.10·13-s + 16-s − 0.727·17-s + 0.229·19-s − 0.377·28-s − 1.11·29-s − 0.718·31-s − 0.328·37-s + 0.937·41-s + 0.152·43-s + 0.904·44-s − 0.437·47-s − 6/7·49-s − 1.10·52-s + 1.64·53-s + 0.781·59-s − 0.128·61-s − 64-s + 0.488·67-s + 0.727·68-s − 0.712·71-s + 0.819·73-s − 0.229·76-s − 0.341·77-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4275 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4275\)    =    \(3^{2} \cdot 5^{2} \cdot 19\)
Sign: $-1$
Analytic conductor: \(34.1360\)
Root analytic conductor: \(5.84260\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{4275} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 4275,\ (\ :1/2),\ -1)\)

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 \( 1 \)
5 \( 1 \)
19 \( 1 - T \)
good2 \( 1 + p T^{2} \)
7 \( 1 - T + p T^{2} \)
11 \( 1 + 3 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - T + p T^{2} \)
47 \( 1 + 3 T + p T^{2} \)
53 \( 1 - 12 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 + T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 - 7 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 12 T + p T^{2} \)
97 \( 1 + 8 T + p T^{2} \)
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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

−8.119095666179137364472777237002, −7.49798376596251597554397220643, −6.51369974795999795736995570396, −5.54613367347237377002063134384, −5.17199046098683380620115379184, −4.14903220539879145832106272998, −3.62000165820744285556163438193, −2.45040466975635997887579513137, −1.28001607419382822143833582519, 0, 1.28001607419382822143833582519, 2.45040466975635997887579513137, 3.62000165820744285556163438193, 4.14903220539879145832106272998, 5.17199046098683380620115379184, 5.54613367347237377002063134384, 6.51369974795999795736995570396, 7.49798376596251597554397220643, 8.119095666179137364472777237002

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