Properties

Label 2-3528-7.2-c1-0-44
Degree $2$
Conductor $3528$
Sign $-0.605 + 0.795i$
Analytic cond. $28.1712$
Root an. cond. $5.30765$
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  + (2.13 − 3.70i)5-s + (−2.13 − 3.70i)11-s + 1.27·13-s + (2 + 3.46i)17-s + (0.637 − 1.10i)19-s + (2 − 3.46i)23-s + (−6.63 − 11.4i)25-s + 2.27·29-s + (−0.5 − 0.866i)31-s + (−2.63 + 4.56i)37-s + 10.5·41-s − 7.27·43-s + (3 − 5.19i)47-s + (0.862 + 1.49i)53-s − 18.2·55-s + ⋯
L(s)  = 1  + (0.955 − 1.65i)5-s + (−0.644 − 1.11i)11-s + 0.353·13-s + (0.485 + 0.840i)17-s + (0.146 − 0.253i)19-s + (0.417 − 0.722i)23-s + (−1.32 − 2.29i)25-s + 0.422·29-s + (−0.0898 − 0.155i)31-s + (−0.433 + 0.751i)37-s + 1.64·41-s − 1.10·43-s + (0.437 − 0.757i)47-s + (0.118 + 0.205i)53-s − 2.46·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.605 + 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3528\)    =    \(2^{3} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.605 + 0.795i$
Analytic conductor: \(28.1712\)
Root analytic conductor: \(5.30765\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3528} (3313, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3528,\ (\ :1/2),\ -0.605 + 0.795i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.010293424\)
\(L(\frac12)\) \(\approx\) \(2.010293424\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good5 \( 1 + (-2.13 + 3.70i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (2.13 + 3.70i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 1.27T + 13T^{2} \)
17 \( 1 + (-2 - 3.46i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.637 + 1.10i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2 + 3.46i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 2.27T + 29T^{2} \)
31 \( 1 + (0.5 + 0.866i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.63 - 4.56i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 10.5T + 41T^{2} \)
43 \( 1 + 7.27T + 43T^{2} \)
47 \( 1 + (-3 + 5.19i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.862 - 1.49i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (3.13 + 5.43i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5 + 8.66i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.63 + 6.30i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 2T + 71T^{2} \)
73 \( 1 + (-1.63 - 2.83i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-1.77 + 3.07i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 0.274T + 83T^{2} \)
89 \( 1 + (2.27 - 3.94i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 16.2T + 97T^{2} \)
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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

−8.358902714264878449229224135977, −7.963201493619097390429281527423, −6.57355034585110333820379251703, −5.91446325531407474128479664930, −5.30964477141230028533100347988, −4.68321397039755359850745506829, −3.66736694168534495385286422734, −2.54628300369273177746442245964, −1.46132342779507892807892236127, −0.59850780740861385076229546921, 1.53527858045962493776003662470, 2.54293324283844981834855025050, 3.04221927774372943704200864024, 4.12581092191666211063382942849, 5.30945524856938289681403299200, 5.80247098895980610771990663890, 6.74818544859454136657252583532, 7.26364983291668438144621071785, 7.77808152570270492178023830326, 9.058181510885200844836081001672

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