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Properties

Label 2-435-145.102-c1-0-4
Degree $2$
Conductor $435$
Sign $0.996 + 0.0868i$
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  + (−1.01 − 0.806i)2-s + (−0.222 + 0.974i)3-s + (−0.0726 − 0.318i)4-s + (−1.73 − 1.41i)5-s + (1.01 − 0.806i)6-s + (−2.96 + 1.86i)7-s + (−1.30 + 2.71i)8-s + (−0.900 − 0.433i)9-s + (0.610 + 2.82i)10-s + (4.30 + 1.50i)11-s + 0.326·12-s + (2.94 + 1.03i)13-s + (4.49 + 0.506i)14-s + (1.76 − 1.37i)15-s + (2.91 − 1.40i)16-s − 7.34i·17-s + ⋯
L(s)  = 1  + (−0.715 − 0.570i)2-s + (−0.128 + 0.562i)3-s + (−0.0363 − 0.159i)4-s + (−0.774 − 0.632i)5-s + (0.412 − 0.329i)6-s + (−1.12 + 0.703i)7-s + (−0.461 + 0.958i)8-s + (−0.300 − 0.144i)9-s + (0.193 + 0.894i)10-s + (1.29 + 0.454i)11-s + 0.0942·12-s + (0.818 + 0.286i)13-s + (1.20 + 0.135i)14-s + (0.455 − 0.354i)15-s + (0.729 − 0.351i)16-s − 1.78i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 + 0.0868i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.996 + 0.0868i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(435\)    =    \(3 \cdot 5 \cdot 29\)
Sign: $0.996 + 0.0868i$
Analytic conductor: \(3.47349\)
Root analytic conductor: \(1.86373\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{435} (247, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 435,\ (\ :1/2),\ 0.996 + 0.0868i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.682394 - 0.0296809i\)
\(L(\frac12)\) \(\approx\) \(0.682394 - 0.0296809i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.222 - 0.974i)T \)
5 \( 1 + (1.73 + 1.41i)T \)
29 \( 1 + (-3.56 + 4.03i)T \)
good2 \( 1 + (1.01 + 0.806i)T + (0.445 + 1.94i)T^{2} \)
7 \( 1 + (2.96 - 1.86i)T + (3.03 - 6.30i)T^{2} \)
11 \( 1 + (-4.30 - 1.50i)T + (8.60 + 6.85i)T^{2} \)
13 \( 1 + (-2.94 - 1.03i)T + (10.1 + 8.10i)T^{2} \)
17 \( 1 + 7.34iT - 17T^{2} \)
19 \( 1 + (-7.34 - 4.61i)T + (8.24 + 17.1i)T^{2} \)
23 \( 1 + (2.83 + 0.319i)T + (22.4 + 5.11i)T^{2} \)
31 \( 1 + (2.56 - 0.289i)T + (30.2 - 6.89i)T^{2} \)
37 \( 1 + (-2.16 - 1.04i)T + (23.0 + 28.9i)T^{2} \)
41 \( 1 + (-1.02 + 1.02i)T - 41iT^{2} \)
43 \( 1 + (-6.71 - 8.41i)T + (-9.56 + 41.9i)T^{2} \)
47 \( 1 + (1.48 - 0.717i)T + (29.3 - 36.7i)T^{2} \)
53 \( 1 + (-1.03 - 9.15i)T + (-51.6 + 11.7i)T^{2} \)
59 \( 1 - 5.26iT - 59T^{2} \)
61 \( 1 + (-2.95 + 1.85i)T + (26.4 - 54.9i)T^{2} \)
67 \( 1 + (-1.61 + 0.565i)T + (52.3 - 41.7i)T^{2} \)
71 \( 1 + (-2.50 - 5.21i)T + (-44.2 + 55.5i)T^{2} \)
73 \( 1 + (-2.64 + 2.10i)T + (16.2 - 71.1i)T^{2} \)
79 \( 1 + (6.21 - 2.17i)T + (61.7 - 49.2i)T^{2} \)
83 \( 1 + (-12.9 - 8.11i)T + (36.0 + 74.7i)T^{2} \)
89 \( 1 + (0.663 + 5.89i)T + (-86.7 + 19.8i)T^{2} \)
97 \( 1 + (-2.29 - 10.0i)T + (-87.3 + 42.0i)T^{2} \)
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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

−11.26311310401370525131349471182, −9.851137539080215937851616648010, −9.422231739551016733627316227097, −8.938579593721907588304895003835, −7.72171718714232037563347740023, −6.31392287607449597667598589308, −5.37115083463936301507680344055, −4.11187790813832812710876022426, −2.96049433477573023494131066366, −1.03420909586196436672030533221, 0.77307159473631679681489060558, 3.40684364144982076730562836904, 3.78347023144170653454353395347, 6.10826952760700630615326915634, 6.70340241087846672369078559162, 7.37430125586133399112942322919, 8.306775992300551551281548113156, 9.113730125597579438596352471401, 10.16856198953398775563290604659, 11.11696041449492555936107015452

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