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Properties

Label 2-3528-7.2-c1-0-24
Degree $2$
Conductor $3528$
Sign $0.991 + 0.126i$
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  + (1 − 1.73i)5-s − 2·13-s + (3 + 5.19i)17-s + (2 − 3.46i)19-s + (−2 + 3.46i)23-s + (0.500 + 0.866i)25-s − 6·29-s + (4 + 6.92i)31-s + (5 − 8.66i)37-s + 10·41-s + 12·43-s + (−4 + 6.92i)47-s + (3 + 5.19i)53-s + (2 + 3.46i)59-s + (5 − 8.66i)61-s + ⋯
L(s)  = 1  + (0.447 − 0.774i)5-s − 0.554·13-s + (0.727 + 1.26i)17-s + (0.458 − 0.794i)19-s + (−0.417 + 0.722i)23-s + (0.100 + 0.173i)25-s − 1.11·29-s + (0.718 + 1.24i)31-s + (0.821 − 1.42i)37-s + 1.56·41-s + 1.82·43-s + (−0.583 + 1.01i)47-s + (0.412 + 0.713i)53-s + (0.260 + 0.450i)59-s + (0.640 − 1.10i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 + 0.126i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 + 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3528\)    =    \(2^{3} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.991 + 0.126i$
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: \((2,\ 3528,\ (\ :1/2),\ 0.991 + 0.126i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.054709959\)
\(L(\frac12)\) \(\approx\) \(2.054709959\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1 + 1.73i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 2T + 13T^{2} \)
17 \( 1 + (-3 - 5.19i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2 + 3.46i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (2 - 3.46i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 6T + 29T^{2} \)
31 \( 1 + (-4 - 6.92i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-5 + 8.66i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 10T + 41T^{2} \)
43 \( 1 - 12T + 43T^{2} \)
47 \( 1 + (4 - 6.92i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-3 - 5.19i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2 - 3.46i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5 + 8.66i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (6 + 10.3i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 4T + 71T^{2} \)
73 \( 1 + (1 + 1.73i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (4 - 6.92i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 4T + 83T^{2} \)
89 \( 1 + (-3 + 5.19i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 10T + 97T^{2} \)
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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

−8.678355668756092883105036180071, −7.69686573983683101313902698306, −7.31504875348357758123373772100, −6.02613233175959143762993001913, −5.66656881112833724196963872937, −4.77122171053755907401490523191, −3.99665905324501884866276771816, −2.96058864872702659600054619445, −1.87780399550877693372696213122, −0.912320271006495895192630618682, 0.813847327531604889499367665169, 2.30345144434421829697701267760, 2.81922683282480476876951518483, 3.90288006719538502716872330338, 4.78831778863037336707564619538, 5.76280529475132472950703505109, 6.22444304302190242756815919464, 7.29173059876157944668133812008, 7.60258066342268017206456313319, 8.563713325925586368755673513647

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