Properties

Label 2-435-1.1-c1-0-7
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.56·2-s + 3-s + 0.438·4-s + 5-s − 1.56·6-s + 5.12·7-s + 2.43·8-s + 9-s − 1.56·10-s − 1.43·11-s + 0.438·12-s − 2·13-s − 8·14-s + 15-s − 4.68·16-s − 7.12·17-s − 1.56·18-s + 5.12·19-s + 0.438·20-s + 5.12·21-s + 2.24·22-s + 6.56·23-s + 2.43·24-s + 25-s + 3.12·26-s + 27-s + 2.24·28-s + ⋯
L(s)  = 1  − 1.10·2-s + 0.577·3-s + 0.219·4-s + 0.447·5-s − 0.637·6-s + 1.93·7-s + 0.862·8-s + 0.333·9-s − 0.493·10-s − 0.433·11-s + 0.126·12-s − 0.554·13-s − 2.13·14-s + 0.258·15-s − 1.17·16-s − 1.72·17-s − 0.368·18-s + 1.17·19-s + 0.0980·20-s + 1.11·21-s + 0.478·22-s + 1.36·23-s + 0.497·24-s + 0.200·25-s + 0.612·26-s + 0.192·27-s + 0.424·28-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: \(0\)
Selberg data: \((2,\ 435,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.160201666\)
\(L(\frac12)\) \(\approx\) \(1.160201666\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.56T + 2T^{2} \)
7 \( 1 - 5.12T + 7T^{2} \)
11 \( 1 + 1.43T + 11T^{2} \)
13 \( 1 + 2T + 13T^{2} \)
17 \( 1 + 7.12T + 17T^{2} \)
19 \( 1 - 5.12T + 19T^{2} \)
23 \( 1 - 6.56T + 23T^{2} \)
31 \( 1 - 4T + 31T^{2} \)
37 \( 1 + 1.68T + 37T^{2} \)
41 \( 1 + 1.68T + 41T^{2} \)
43 \( 1 - 7.68T + 43T^{2} \)
47 \( 1 + 13.1T + 47T^{2} \)
53 \( 1 + 3.43T + 53T^{2} \)
59 \( 1 - 12T + 59T^{2} \)
61 \( 1 - 0.876T + 61T^{2} \)
67 \( 1 + 11.3T + 67T^{2} \)
71 \( 1 + 2.87T + 71T^{2} \)
73 \( 1 - 1.68T + 73T^{2} \)
79 \( 1 + 12T + 79T^{2} \)
83 \( 1 + 2.56T + 83T^{2} \)
89 \( 1 - 12.2T + 89T^{2} \)
97 \( 1 + 5.68T + 97T^{2} \)
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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

−10.96417206488600490263655285433, −10.10904995326953672334215539315, −9.118052207317669976865804380000, −8.543204798214796364896863540620, −7.75235415881385980632209288744, −7.00095725827243836204529869390, −5.11028811136306908507362470100, −4.54100118367316077581297147597, −2.46123361879166738208870635262, −1.36360332662919032380159427521, 1.36360332662919032380159427521, 2.46123361879166738208870635262, 4.54100118367316077581297147597, 5.11028811136306908507362470100, 7.00095725827243836204529869390, 7.75235415881385980632209288744, 8.543204798214796364896863540620, 9.118052207317669976865804380000, 10.10904995326953672334215539315, 10.96417206488600490263655285433

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