Properties

Label 2-435-1.1-c1-0-8
Degree $2$
Conductor $435$
Sign $-1$
Analytic cond. $3.47349$
Root an. cond. $1.86373$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.75·2-s − 3-s + 5.59·4-s − 5-s + 2.75·6-s + 0.393·7-s − 9.92·8-s + 9-s + 2.75·10-s − 0.393·11-s − 5.59·12-s − 2.56·13-s − 1.08·14-s + 15-s + 16.1·16-s + 2.07·17-s − 2.75·18-s − 0.958·19-s − 5.59·20-s − 0.393·21-s + 1.08·22-s + 6.15·23-s + 9.92·24-s + 25-s + 7.07·26-s − 27-s + 2.20·28-s + ⋯
L(s)  = 1  − 1.94·2-s − 0.577·3-s + 2.79·4-s − 0.447·5-s + 1.12·6-s + 0.148·7-s − 3.50·8-s + 0.333·9-s + 0.871·10-s − 0.118·11-s − 1.61·12-s − 0.711·13-s − 0.290·14-s + 0.258·15-s + 4.03·16-s + 0.504·17-s − 0.649·18-s − 0.219·19-s − 1.25·20-s − 0.0859·21-s + 0.231·22-s + 1.28·23-s + 2.02·24-s + 0.200·25-s + 1.38·26-s − 0.192·27-s + 0.416·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(435\)    =    \(3 \cdot 5 \cdot 29\)
Sign: $-1$
Analytic conductor: \(3.47349\)
Root analytic conductor: \(1.86373\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 435,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
29 \( 1 + T \)
good2 \( 1 + 2.75T + 2T^{2} \)
7 \( 1 - 0.393T + 7T^{2} \)
11 \( 1 + 0.393T + 11T^{2} \)
13 \( 1 + 2.56T + 13T^{2} \)
17 \( 1 - 2.07T + 17T^{2} \)
19 \( 1 + 0.958T + 19T^{2} \)
23 \( 1 - 6.15T + 23T^{2} \)
31 \( 1 + 10.1T + 31T^{2} \)
37 \( 1 + 7.34T + 37T^{2} \)
41 \( 1 + 1.65T + 41T^{2} \)
43 \( 1 - 10.3T + 43T^{2} \)
47 \( 1 + 11.5T + 47T^{2} \)
53 \( 1 + 12.3T + 53T^{2} \)
59 \( 1 + 9.54T + 59T^{2} \)
61 \( 1 + 6.25T + 61T^{2} \)
67 \( 1 + 7.42T + 67T^{2} \)
71 \( 1 - 5.98T + 71T^{2} \)
73 \( 1 - 3.34T + 73T^{2} \)
79 \( 1 + 2.06T + 79T^{2} \)
83 \( 1 - 6.41T + 83T^{2} \)
89 \( 1 - 15.8T + 89T^{2} \)
97 \( 1 + 18.4T + 97T^{2} \)
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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

−10.78915568034698335277944813985, −9.662452912036520907773497197686, −9.031655652508442904977254264482, −7.906494491978344271111571696771, −7.31937622994003106393735664024, −6.42977435960343359446140168654, −5.17345265331665152968490762031, −3.17707041902984177898901916665, −1.62177486373104229396777593335, 0, 1.62177486373104229396777593335, 3.17707041902984177898901916665, 5.17345265331665152968490762031, 6.42977435960343359446140168654, 7.31937622994003106393735664024, 7.906494491978344271111571696771, 9.031655652508442904977254264482, 9.662452912036520907773497197686, 10.78915568034698335277944813985

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