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Properties

Label 2-435-145.99-c2-0-36
Degree $2$
Conductor $435$
Sign $0.828 + 0.560i$
Analytic cond. $11.8528$
Root an. cond. $3.44280$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.22 − 1.22i)2-s + (1.22 − 1.22i)3-s + 1.00i·4-s − 5i·5-s − 2.99i·6-s + 9.79i·7-s + (6.12 + 6.12i)8-s − 2.99i·9-s + (−6.12 − 6.12i)10-s + (7 + 7i)11-s + (1.22 + 1.22i)12-s + 9.79·13-s + (11.9 + 11.9i)14-s + (−6.12 − 6.12i)15-s + 10.9·16-s + (14.6 − 14.6i)17-s + ⋯
L(s)  = 1  + (0.612 − 0.612i)2-s + (0.408 − 0.408i)3-s + 0.250i·4-s − i·5-s − 0.499i·6-s + 1.39i·7-s + (0.765 + 0.765i)8-s − 0.333i·9-s + (−0.612 − 0.612i)10-s + (0.636 + 0.636i)11-s + (0.102 + 0.102i)12-s + 0.753·13-s + (0.857 + 0.857i)14-s + (−0.408 − 0.408i)15-s + 0.687·16-s + (0.864 − 0.864i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.828 + 0.560i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.828 + 0.560i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(435\)    =    \(3 \cdot 5 \cdot 29\)
Sign: $0.828 + 0.560i$
Analytic conductor: \(11.8528\)
Root analytic conductor: \(3.44280\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{435} (244, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 435,\ (\ :1),\ 0.828 + 0.560i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.90206 - 0.889999i\)
\(L(\frac12)\) \(\approx\) \(2.90206 - 0.889999i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.22 + 1.22i)T \)
5 \( 1 + 5iT \)
29 \( 1 - 29iT \)
good2 \( 1 + (-1.22 + 1.22i)T - 4iT^{2} \)
7 \( 1 - 9.79iT - 49T^{2} \)
11 \( 1 + (-7 - 7i)T + 121iT^{2} \)
13 \( 1 - 9.79T + 169T^{2} \)
17 \( 1 + (-14.6 + 14.6i)T - 289iT^{2} \)
19 \( 1 + (-11 - 11i)T + 361iT^{2} \)
23 \( 1 + 39.1iT - 529T^{2} \)
31 \( 1 + (11 + 11i)T + 961iT^{2} \)
37 \( 1 + (34.2 + 34.2i)T + 1.36e3iT^{2} \)
41 \( 1 + (-5 + 5i)T - 1.68e3iT^{2} \)
43 \( 1 + (34.2 - 34.2i)T - 1.84e3iT^{2} \)
47 \( 1 + (-53.8 - 53.8i)T + 2.20e3iT^{2} \)
53 \( 1 + 88.1iT - 2.80e3T^{2} \)
59 \( 1 + 24T + 3.48e3T^{2} \)
61 \( 1 + (-49 - 49i)T + 3.72e3iT^{2} \)
67 \( 1 - 19.5T + 4.48e3T^{2} \)
71 \( 1 - 130iT - 5.04e3T^{2} \)
73 \( 1 + (4.89 + 4.89i)T + 5.32e3iT^{2} \)
79 \( 1 + (-11 - 11i)T + 6.24e3iT^{2} \)
83 \( 1 + 117. iT - 6.88e3T^{2} \)
89 \( 1 + (77 + 77i)T + 7.92e3iT^{2} \)
97 \( 1 + (-14.6 - 14.6i)T + 9.40e3iT^{2} \)
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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

−11.29376845578516188831655266211, −9.821477649070056540865056666657, −8.830892639024053761590531130146, −8.387526286979838320869673354165, −7.25011011944195766385822760988, −5.80895458973012653465059955251, −4.92924564377723670456751790235, −3.77381982084722598759183707908, −2.59909292680263386034991877224, −1.48359906650295556374639677992, 1.29717919273442020583075745683, 3.56824094425341047681114819615, 3.83511200390625104979612462571, 5.36258450961777251644687129070, 6.33938678987709140143129518937, 7.16972457557311037477405398086, 7.930757446833597072952449995641, 9.386912493989578982492968346191, 10.29313852994934632793163936454, 10.75431138901599506833127905657

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