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Properties

Label 2-3528-7.4-c1-0-31
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  + (0.5 + 0.866i)5-s + (1.5 − 2.59i)11-s − 4·13-s + (−2 − 3.46i)19-s + (4 + 6.92i)23-s + (2 − 3.46i)25-s + 3·29-s + (−2.5 + 4.33i)31-s + (−4 − 6.92i)37-s + 8·41-s + 6·43-s + (−5 − 8.66i)47-s + (4.5 − 7.79i)53-s + 3·55-s + (2.5 − 4.33i)59-s + ⋯
L(s)  = 1  + (0.223 + 0.387i)5-s + (0.452 − 0.783i)11-s − 1.10·13-s + (−0.458 − 0.794i)19-s + (0.834 + 1.44i)23-s + (0.400 − 0.692i)25-s + 0.557·29-s + (−0.449 + 0.777i)31-s + (−0.657 − 1.13i)37-s + 1.24·41-s + 0.914·43-s + (−0.729 − 1.26i)47-s + (0.618 − 1.07i)53-s + 0.404·55-s + (0.325 − 0.563i)59-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: $\chi_{3528} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3528,\ (\ :1/2),\ 0.605 + 0.795i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.655014260\)
\(L(\frac12)\) \(\approx\) \(1.655014260\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 - 0.866i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.5 + 2.59i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 4T + 13T^{2} \)
17 \( 1 + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2 + 3.46i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-4 - 6.92i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 3T + 29T^{2} \)
31 \( 1 + (2.5 - 4.33i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (4 + 6.92i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 8T + 41T^{2} \)
43 \( 1 - 6T + 43T^{2} \)
47 \( 1 + (5 + 8.66i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-4.5 + 7.79i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2.5 + 4.33i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5 + 8.66i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3 - 5.19i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 10T + 71T^{2} \)
73 \( 1 + (-1 + 1.73i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (5.5 + 9.52i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 7T + 83T^{2} \)
89 \( 1 + (-9 - 15.5i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 17T + 97T^{2} \)
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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

−8.629227630009887279145540022626, −7.55292415496206884369343412541, −7.02407493871765912489542844214, −6.29485844519005749814456139549, −5.41586058684899082141644201121, −4.74086892479543541939111575090, −3.66592370656790317181472622958, −2.88500172664999356380903062121, −1.95834232860131835420589245401, −0.55554333174917070613506621137, 1.06611658159492471904380735698, 2.17012861559638173780064386965, 3.01980056397732353003577921481, 4.37071512007147997793959811230, 4.65388339232225585497341646193, 5.68939193425972804540635666383, 6.45845115491488280649068569645, 7.26413396495480902480192714257, 7.83301046189338118479330706563, 8.906452039533697385749912433495

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