Properties

Label 2-2175-1.1-c1-0-27
Degree $2$
Conductor $2175$
Sign $1$
Analytic cond. $17.3674$
Root an. cond. $4.16742$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.79·2-s − 3-s + 1.20·4-s + 1.79·6-s − 7-s + 1.41·8-s + 9-s + 5·11-s − 1.20·12-s + 4.58·13-s + 1.79·14-s − 4.95·16-s + 3·17-s − 1.79·18-s + 3.58·19-s + 21-s − 8.95·22-s + 4·23-s − 1.41·24-s − 8.20·26-s − 27-s − 1.20·28-s + 29-s + 4·31-s + 6.04·32-s − 5·33-s − 5.37·34-s + ⋯
L(s)  = 1  − 1.26·2-s − 0.577·3-s + 0.604·4-s + 0.731·6-s − 0.377·7-s + 0.501·8-s + 0.333·9-s + 1.50·11-s − 0.348·12-s + 1.27·13-s + 0.478·14-s − 1.23·16-s + 0.727·17-s − 0.422·18-s + 0.821·19-s + 0.218·21-s − 1.90·22-s + 0.834·23-s − 0.289·24-s − 1.60·26-s − 0.192·27-s − 0.228·28-s + 0.185·29-s + 0.718·31-s + 1.06·32-s − 0.870·33-s − 0.921·34-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: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2175,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8845952162\)
\(L(\frac12)\) \(\approx\) \(0.8845952162\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.79T + 2T^{2} \)
7 \( 1 + T + 7T^{2} \)
11 \( 1 - 5T + 11T^{2} \)
13 \( 1 - 4.58T + 13T^{2} \)
17 \( 1 - 3T + 17T^{2} \)
19 \( 1 - 3.58T + 19T^{2} \)
23 \( 1 - 4T + 23T^{2} \)
31 \( 1 - 4T + 31T^{2} \)
37 \( 1 - 4T + 37T^{2} \)
41 \( 1 + 9.16T + 41T^{2} \)
43 \( 1 - 9.58T + 43T^{2} \)
47 \( 1 + 10.5T + 47T^{2} \)
53 \( 1 + 0.417T + 53T^{2} \)
59 \( 1 + 7.58T + 59T^{2} \)
61 \( 1 - 12.7T + 61T^{2} \)
67 \( 1 - 4.16T + 67T^{2} \)
71 \( 1 + 9.58T + 71T^{2} \)
73 \( 1 + 4T + 73T^{2} \)
79 \( 1 - 7.58T + 79T^{2} \)
83 \( 1 - 11.5T + 83T^{2} \)
89 \( 1 - 1.41T + 89T^{2} \)
97 \( 1 + 11.5T + 97T^{2} \)
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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

−9.162116754132302553154066006700, −8.442482924588290781486228239667, −7.65815231598995003054480680296, −6.71364743734635390796179695737, −6.30162114434693409334273495843, −5.17675596705504956908515818125, −4.13960371082335566514743268559, −3.23214326805480175550137597333, −1.49958259453774684915816459667, −0.881935906972457734636824661786, 0.881935906972457734636824661786, 1.49958259453774684915816459667, 3.23214326805480175550137597333, 4.13960371082335566514743268559, 5.17675596705504956908515818125, 6.30162114434693409334273495843, 6.71364743734635390796179695737, 7.65815231598995003054480680296, 8.442482924588290781486228239667, 9.162116754132302553154066006700

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