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Properties

Label 2-3528-7.4-c1-0-5
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 + (−2.5 + 4.33i)11-s − 2·13-s + (3 − 5.19i)17-s + (1 + 1.73i)19-s + (3 + 5.19i)23-s + (2 − 3.46i)25-s − 3·29-s + (2.5 − 4.33i)31-s + (1 + 1.73i)37-s − 8·41-s − 4·43-s + (2 + 3.46i)47-s + (−4.5 + 7.79i)53-s + 5·55-s + ⋯
L(s)  = 1  + (−0.223 − 0.387i)5-s + (−0.753 + 1.30i)11-s − 0.554·13-s + (0.727 − 1.26i)17-s + (0.229 + 0.397i)19-s + (0.625 + 1.08i)23-s + (0.400 − 0.692i)25-s − 0.557·29-s + (0.449 − 0.777i)31-s + (0.164 + 0.284i)37-s − 1.24·41-s − 0.609·43-s + (0.291 + 0.505i)47-s + (−0.618 + 1.07i)53-s + 0.674·55-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: $\chi_{3528} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3528,\ (\ :1/2),\ -0.605 - 0.795i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6728971590\)
\(L(\frac12)\) \(\approx\) \(0.6728971590\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.5 - 4.33i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 2T + 13T^{2} \)
17 \( 1 + (-3 + 5.19i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1 - 1.73i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3 - 5.19i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 3T + 29T^{2} \)
31 \( 1 + (-2.5 + 4.33i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1 - 1.73i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 8T + 41T^{2} \)
43 \( 1 + 4T + 43T^{2} \)
47 \( 1 + (-2 - 3.46i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (4.5 - 7.79i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.5 - 2.59i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (6 + 10.3i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1 - 1.73i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 8T + 71T^{2} \)
73 \( 1 + (7 - 12.1i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.5 + 0.866i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 17T + 83T^{2} \)
89 \( 1 + (-9 - 15.5i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 3T + 97T^{2} \)
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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

−8.880270629355251418862399527017, −7.83498578057673026619539977767, −7.52669261002076488952534918156, −6.78983931557457659385539245081, −5.64071863429174267870408098164, −4.97344933325940742705700055305, −4.44308033687073639549303243163, −3.24513434398377827482029622112, −2.42951162489940619478945684015, −1.26008085886657065669423903893, 0.20618910303543904666565201831, 1.60296791239503275272279051742, 2.98400554326857043810168306977, 3.29936764299892542661966157190, 4.51064034597458007719311144849, 5.35208530430510362765667495339, 6.01693926606808196310259257060, 6.89294359364281671184791953893, 7.55460132628767990258455822862, 8.443300000770050302740119356379

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