Properties

Label 2-435-1.1-c3-0-3
Degree $2$
Conductor $435$
Sign $1$
Analytic cond. $25.6658$
Root an. cond. $5.06614$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.218·2-s − 3·3-s − 7.95·4-s + 5·5-s − 0.656·6-s − 20.9·7-s − 3.48·8-s + 9·9-s + 1.09·10-s − 1.46·11-s + 23.8·12-s − 79.4·13-s − 4.59·14-s − 15·15-s + 62.8·16-s − 65.4·17-s + 1.96·18-s + 116.·19-s − 39.7·20-s + 62.9·21-s − 0.320·22-s − 177.·23-s + 10.4·24-s + 25·25-s − 17.3·26-s − 27·27-s + 166.·28-s + ⋯
L(s)  = 1  + 0.0773·2-s − 0.577·3-s − 0.994·4-s + 0.447·5-s − 0.0446·6-s − 1.13·7-s − 0.154·8-s + 0.333·9-s + 0.0345·10-s − 0.0401·11-s + 0.573·12-s − 1.69·13-s − 0.0876·14-s − 0.258·15-s + 0.982·16-s − 0.933·17-s + 0.0257·18-s + 1.41·19-s − 0.444·20-s + 0.654·21-s − 0.00310·22-s − 1.60·23-s + 0.0890·24-s + 0.200·25-s − 0.131·26-s − 0.192·27-s + 1.12·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(435\)    =    \(3 \cdot 5 \cdot 29\)
Sign: $1$
Analytic conductor: \(25.6658\)
Root analytic conductor: \(5.06614\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 435,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(0.7096604796\)
\(L(\frac12)\) \(\approx\) \(0.7096604796\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 3T \)
5 \( 1 - 5T \)
29 \( 1 - 29T \)
good2 \( 1 - 0.218T + 8T^{2} \)
7 \( 1 + 20.9T + 343T^{2} \)
11 \( 1 + 1.46T + 1.33e3T^{2} \)
13 \( 1 + 79.4T + 2.19e3T^{2} \)
17 \( 1 + 65.4T + 4.91e3T^{2} \)
19 \( 1 - 116.T + 6.85e3T^{2} \)
23 \( 1 + 177.T + 1.21e4T^{2} \)
31 \( 1 - 77.7T + 2.97e4T^{2} \)
37 \( 1 - 175.T + 5.06e4T^{2} \)
41 \( 1 - 62.4T + 6.89e4T^{2} \)
43 \( 1 - 66.0T + 7.95e4T^{2} \)
47 \( 1 - 179.T + 1.03e5T^{2} \)
53 \( 1 - 676.T + 1.48e5T^{2} \)
59 \( 1 - 415.T + 2.05e5T^{2} \)
61 \( 1 + 426.T + 2.26e5T^{2} \)
67 \( 1 - 79.1T + 3.00e5T^{2} \)
71 \( 1 - 218.T + 3.57e5T^{2} \)
73 \( 1 - 1.02e3T + 3.89e5T^{2} \)
79 \( 1 - 1.05e3T + 4.93e5T^{2} \)
83 \( 1 + 1.17e3T + 5.71e5T^{2} \)
89 \( 1 - 1.41e3T + 7.04e5T^{2} \)
97 \( 1 - 884.T + 9.12e5T^{2} \)
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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

−10.42745597050537262801924540209, −9.692316285197396707153188638413, −9.357731139814288913072143888818, −7.936837887501491501547013412376, −6.87289691323759682479755572143, −5.85234417985659007857250130803, −5.00982384976872826631653443736, −3.96020834428211944202156491478, −2.54374036600031587190696941934, −0.53059391121560305721812868442, 0.53059391121560305721812868442, 2.54374036600031587190696941934, 3.96020834428211944202156491478, 5.00982384976872826631653443736, 5.85234417985659007857250130803, 6.87289691323759682479755572143, 7.936837887501491501547013412376, 9.357731139814288913072143888818, 9.692316285197396707153188638413, 10.42745597050537262801924540209

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