Properties

Label 2-675-1.1-c3-0-46
Degree $2$
Conductor $675$
Sign $-1$
Analytic cond. $39.8262$
Root an. cond. $6.31080$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3.60·2-s + 4.99·4-s + 9·7-s + 10.8·8-s + 14.4·11-s − 34·13-s − 32.4·14-s − 79·16-s + 7.21·17-s − 101·19-s − 51.9·22-s + 108.·23-s + 122.·26-s + 44.9·28-s − 165.·29-s − 3·31-s + 198.·32-s − 25.9·34-s + 67·37-s + 364.·38-s + 201.·41-s + 137·43-s + 72.1·44-s − 390·46-s + 115.·47-s − 262·49-s − 169.·52-s + ⋯
L(s)  = 1  − 1.27·2-s + 0.624·4-s + 0.485·7-s + 0.478·8-s + 0.395·11-s − 0.725·13-s − 0.619·14-s − 1.23·16-s + 0.102·17-s − 1.21·19-s − 0.503·22-s + 0.980·23-s + 0.924·26-s + 0.303·28-s − 1.06·29-s − 0.0173·31-s + 1.09·32-s − 0.131·34-s + 0.297·37-s + 1.55·38-s + 0.769·41-s + 0.485·43-s + 0.247·44-s − 1.25·46-s + 0.358·47-s − 0.763·49-s − 0.453·52-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(675\)    =    \(3^{3} \cdot 5^{2}\)
Sign: $-1$
Analytic conductor: \(39.8262\)
Root analytic conductor: \(6.31080\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 675,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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 \( 1 \)
good2 \( 1 + 3.60T + 8T^{2} \)
7 \( 1 - 9T + 343T^{2} \)
11 \( 1 - 14.4T + 1.33e3T^{2} \)
13 \( 1 + 34T + 2.19e3T^{2} \)
17 \( 1 - 7.21T + 4.91e3T^{2} \)
19 \( 1 + 101T + 6.85e3T^{2} \)
23 \( 1 - 108.T + 1.21e4T^{2} \)
29 \( 1 + 165.T + 2.43e4T^{2} \)
31 \( 1 + 3T + 2.97e4T^{2} \)
37 \( 1 - 67T + 5.06e4T^{2} \)
41 \( 1 - 201.T + 6.89e4T^{2} \)
43 \( 1 - 137T + 7.95e4T^{2} \)
47 \( 1 - 115.T + 1.03e5T^{2} \)
53 \( 1 - 677.T + 1.48e5T^{2} \)
59 \( 1 + 872.T + 2.05e5T^{2} \)
61 \( 1 + 563T + 2.26e5T^{2} \)
67 \( 1 + 1.04e3T + 3.00e5T^{2} \)
71 \( 1 - 786.T + 3.57e5T^{2} \)
73 \( 1 - 503T + 3.89e5T^{2} \)
79 \( 1 - 615T + 4.93e5T^{2} \)
83 \( 1 + 237.T + 5.71e5T^{2} \)
89 \( 1 - 973.T + 7.04e5T^{2} \)
97 \( 1 - 19T + 9.12e5T^{2} \)
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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

−9.378846801564981211213098478464, −9.023143744690717401032053349877, −7.975935884957555559527193911512, −7.37993672431339052953696958473, −6.37641982864820342821759461366, −5.04310895801708967401223201553, −4.10110347115195155218451467304, −2.43137051438313020062994587431, −1.31521111768489924789725266447, 0, 1.31521111768489924789725266447, 2.43137051438313020062994587431, 4.10110347115195155218451467304, 5.04310895801708967401223201553, 6.37641982864820342821759461366, 7.37993672431339052953696958473, 7.975935884957555559527193911512, 9.023143744690717401032053349877, 9.378846801564981211213098478464

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