# Properties

 Label 2-735-1.1-c3-0-42 Degree $2$ Conductor $735$ Sign $1$ Analytic cond. $43.3664$ Root an. cond. $6.58531$ Motivic weight $3$ Arithmetic yes Rational no Primitive yes Self-dual yes Analytic rank $0$

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## Dirichlet series

 L(s)  = 1 + 4.70·2-s − 3·3-s + 14.1·4-s + 5·5-s − 14.1·6-s + 28.7·8-s + 9·9-s + 23.5·10-s + 24.5·11-s − 42.3·12-s + 35.0·13-s − 15·15-s + 22.1·16-s + 18.4·17-s + 42.3·18-s + 67.4·19-s + 70.5·20-s + 115.·22-s − 145.·23-s − 86.1·24-s + 25·25-s + 164.·26-s − 27·27-s + 214.·29-s − 70.5·30-s + 88.6·31-s − 125.·32-s + ⋯
 L(s)  = 1 + 1.66·2-s − 0.577·3-s + 1.76·4-s + 0.447·5-s − 0.959·6-s + 1.26·8-s + 0.333·9-s + 0.743·10-s + 0.674·11-s − 1.01·12-s + 0.747·13-s − 0.258·15-s + 0.345·16-s + 0.262·17-s + 0.554·18-s + 0.813·19-s + 0.788·20-s + 1.12·22-s − 1.32·23-s − 0.732·24-s + 0.200·25-s + 1.24·26-s − 0.192·27-s + 1.37·29-s − 0.429·30-s + 0.513·31-s − 0.694·32-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$735$$    =    $$3 \cdot 5 \cdot 7^{2}$$ Sign: $1$ Analytic conductor: $$43.3664$$ Root analytic conductor: $$6.58531$$ Motivic weight: $$3$$ Rational: no Arithmetic: yes Character: Trivial Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(2,\ 735,\ (\ :3/2),\ 1)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$5.486027028$$ $$L(\frac12)$$ $$\approx$$ $$5.486027028$$ $$L(\frac{5}{2})$$ 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$$
7 $$1$$
good2 $$1 - 4.70T + 8T^{2}$$
11 $$1 - 24.5T + 1.33e3T^{2}$$
13 $$1 - 35.0T + 2.19e3T^{2}$$
17 $$1 - 18.4T + 4.91e3T^{2}$$
19 $$1 - 67.4T + 6.85e3T^{2}$$
23 $$1 + 145.T + 1.21e4T^{2}$$
29 $$1 - 214.T + 2.43e4T^{2}$$
31 $$1 - 88.6T + 2.97e4T^{2}$$
37 $$1 - 162.T + 5.06e4T^{2}$$
41 $$1 - 337.T + 6.89e4T^{2}$$
43 $$1 - 122.T + 7.95e4T^{2}$$
47 $$1 + 354.T + 1.03e5T^{2}$$
53 $$1 - 676.T + 1.48e5T^{2}$$
59 $$1 + 501.T + 2.05e5T^{2}$$
61 $$1 - 708.T + 2.26e5T^{2}$$
67 $$1 + 907.T + 3.00e5T^{2}$$
71 $$1 - 430.T + 3.57e5T^{2}$$
73 $$1 + 41.3T + 3.89e5T^{2}$$
79 $$1 - 890.T + 4.93e5T^{2}$$
83 $$1 - 1.05e3T + 5.71e5T^{2}$$
89 $$1 + 1.47e3T + 7.04e5T^{2}$$
97 $$1 + 555.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

−10.26243306453489764673215649701, −9.312958750931259854264478827614, −8.014715513445701045886512751649, −6.80558046091901650361741660254, −6.15139409839427904295959889246, −5.53996684522321551081043933933, −4.51515373973666636588781518725, −3.73929958869782030011687353693, −2.55989918564782867568502447585, −1.17920045757108547304782338798, 1.17920045757108547304782338798, 2.55989918564782867568502447585, 3.73929958869782030011687353693, 4.51515373973666636588781518725, 5.53996684522321551081043933933, 6.15139409839427904295959889246, 6.80558046091901650361741660254, 8.014715513445701045886512751649, 9.312958750931259854264478827614, 10.26243306453489764673215649701