↑

Properties

Label 2-4284-17.4-c1-0-2
Degree $2$
Conductor $4284$
Sign $-0.989 + 0.143i$
Analytic cond. $34.2079$
Root an. cond. $5.84875$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−1.25 + 1.25i)5-s + (0.707 + 0.707i)7-s + (2.12 + 2.12i)11-s − 2.59·13-s + (−1.08 − 3.97i)17-s + 4.57i·19-s + (−3.09 − 3.09i)23-s + 1.86i·25-s + (−1.70 + 1.70i)29-s + (1.86 − 1.86i)31-s − 1.77·35-s + (0.372 − 0.372i)37-s + (8.52 + 8.52i)41-s + 2.63i·43-s − 9.67·47-s + ⋯
L(s)  = 1  + (−0.560 + 0.560i)5-s + (0.267 + 0.267i)7-s + (0.639 + 0.639i)11-s − 0.720·13-s + (−0.263 − 0.964i)17-s + 1.04i·19-s + (−0.644 − 0.644i)23-s + 0.372i·25-s + (−0.316 + 0.316i)29-s + (0.335 − 0.335i)31-s − 0.299·35-s + (0.0612 − 0.0612i)37-s + (1.33 + 1.33i)41-s + 0.402i·43-s − 1.41·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4284 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.989 + 0.143i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4284 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.989 + 0.143i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4284\)    =    \(2^{2} \cdot 3^{2} \cdot 7 \cdot 17\)
Sign: $-0.989 + 0.143i$
Analytic conductor: \(34.2079\)
Root analytic conductor: \(5.84875\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4284} (4033, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4284,\ (\ :1/2),\ -0.989 + 0.143i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4122573389\)
\(L(\frac12)\) \(\approx\) \(0.4122573389\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 + (-0.707 - 0.707i)T \)
17 \( 1 + (1.08 + 3.97i)T \)
good5 \( 1 + (1.25 - 1.25i)T - 5iT^{2} \)
11 \( 1 + (-2.12 - 2.12i)T + 11iT^{2} \)
13 \( 1 + 2.59T + 13T^{2} \)
19 \( 1 - 4.57iT - 19T^{2} \)
23 \( 1 + (3.09 + 3.09i)T + 23iT^{2} \)
29 \( 1 + (1.70 - 1.70i)T - 29iT^{2} \)
31 \( 1 + (-1.86 + 1.86i)T - 31iT^{2} \)
37 \( 1 + (-0.372 + 0.372i)T - 37iT^{2} \)
41 \( 1 + (-8.52 - 8.52i)T + 41iT^{2} \)
43 \( 1 - 2.63iT - 43T^{2} \)
47 \( 1 + 9.67T + 47T^{2} \)
53 \( 1 + 3.88iT - 53T^{2} \)
59 \( 1 + 2.63iT - 59T^{2} \)
61 \( 1 + (6.61 + 6.61i)T + 61iT^{2} \)
67 \( 1 + 14.9T + 67T^{2} \)
71 \( 1 + (10.1 - 10.1i)T - 71iT^{2} \)
73 \( 1 + (-2.22 + 2.22i)T - 73iT^{2} \)
79 \( 1 + (-12.2 - 12.2i)T + 79iT^{2} \)
83 \( 1 + 12.6iT - 83T^{2} \)
89 \( 1 + 2.05T + 89T^{2} \)
97 \( 1 + (-5.08 + 5.08i)T - 97iT^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.721258524622026226992200826183, −7.83421434268766365614337477670, −7.44318492294681857180512084483, −6.62715173944931811348381365012, −5.94987892155813546545957563547, −4.87329171354037681969327420960, −4.33110345133497116660275100775, −3.36642470547638034309603468408, −2.52240391368170323906100751277, −1.51684071757675467573098975644, 0.11904979256340121145620600998, 1.26575016074157810792368210388, 2.39310618181674495304608371329, 3.53138834227532555850093100977, 4.25246153816916240512087238170, 4.84741068933841477436032792791, 5.84432913833350252015971263355, 6.50275231233782144783337377743, 7.48814984022643412903506029917, 7.906996605391901355830194400859

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