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Properties

Label 2-435-87.17-c1-0-30
Degree $2$
Conductor $435$
Sign $0.316 + 0.948i$
Analytic cond. $3.47349$
Root an. cond. $1.86373$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.0236 − 0.0236i)2-s + (1.03 − 1.39i)3-s − 1.99i·4-s + 5-s + (−0.0572 + 0.00855i)6-s + 3.42·7-s + (−0.0944 + 0.0944i)8-s + (−0.877 − 2.86i)9-s + (−0.0236 − 0.0236i)10-s + (4.10 + 4.10i)11-s + (−2.78 − 2.05i)12-s + 3.19i·13-s + (−0.0809 − 0.0809i)14-s + (1.03 − 1.39i)15-s − 3.99·16-s + (−4.08 − 4.08i)17-s + ⋯
L(s)  = 1  + (−0.0167 − 0.0167i)2-s + (0.594 − 0.803i)3-s − 0.999i·4-s + 0.447·5-s + (−0.0233 + 0.00349i)6-s + 1.29·7-s + (−0.0334 + 0.0334i)8-s + (−0.292 − 0.956i)9-s + (−0.00747 − 0.00747i)10-s + (1.23 + 1.23i)11-s + (−0.803 − 0.594i)12-s + 0.886i·13-s + (−0.0216 − 0.0216i)14-s + (0.266 − 0.359i)15-s − 0.998·16-s + (−0.991 − 0.991i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.316 + 0.948i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.316 + 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(435\)    =    \(3 \cdot 5 \cdot 29\)
Sign: $0.316 + 0.948i$
Analytic conductor: \(3.47349\)
Root analytic conductor: \(1.86373\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{435} (191, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 435,\ (\ :1/2),\ 0.316 + 0.948i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.61426 - 1.16296i\)
\(L(\frac12)\) \(\approx\) \(1.61426 - 1.16296i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.03 + 1.39i)T \)
5 \( 1 - T \)
29 \( 1 + (4.91 + 2.20i)T \)
good2 \( 1 + (0.0236 + 0.0236i)T + 2iT^{2} \)
7 \( 1 - 3.42T + 7T^{2} \)
11 \( 1 + (-4.10 - 4.10i)T + 11iT^{2} \)
13 \( 1 - 3.19iT - 13T^{2} \)
17 \( 1 + (4.08 + 4.08i)T + 17iT^{2} \)
19 \( 1 + (1.51 - 1.51i)T - 19iT^{2} \)
23 \( 1 - 3.49iT - 23T^{2} \)
31 \( 1 + (0.896 - 0.896i)T - 31iT^{2} \)
37 \( 1 + (1.53 + 1.53i)T + 37iT^{2} \)
41 \( 1 + (4.78 - 4.78i)T - 41iT^{2} \)
43 \( 1 + (2.79 - 2.79i)T - 43iT^{2} \)
47 \( 1 + (3.94 - 3.94i)T - 47iT^{2} \)
53 \( 1 - 0.219iT - 53T^{2} \)
59 \( 1 + 14.4iT - 59T^{2} \)
61 \( 1 + (-7.78 + 7.78i)T - 61iT^{2} \)
67 \( 1 - 11.2iT - 67T^{2} \)
71 \( 1 - 9.21T + 71T^{2} \)
73 \( 1 + (-10.7 - 10.7i)T + 73iT^{2} \)
79 \( 1 + (-1.49 + 1.49i)T - 79iT^{2} \)
83 \( 1 + 1.76iT - 83T^{2} \)
89 \( 1 + (6.11 + 6.11i)T + 89iT^{2} \)
97 \( 1 + (5.10 + 5.10i)T + 97iT^{2} \)
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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

−11.33893424911171949303701743203, −9.650677208572663039214063668349, −9.370291753436988581513591487130, −8.285993568126358370207538971472, −7.02690318245873385884596682738, −6.54999165561739082231107041827, −5.17752174190738397096689283095, −4.20686821401400151111898036903, −2.02772973646363139696817925651, −1.58465699320462925997243846950, 2.07744353629106760757838764030, 3.44792478840971265213192353727, 4.26910324930048497313111029370, 5.40969345099680155710723874315, 6.74511877117308423294238787883, 8.133990276728315148641922961757, 8.528139452603041069370723952259, 9.187712099158887929594400641064, 10.68260013104685804111311797662, 11.07497149854108421725057190019

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