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Properties

Label 2-3528-7.4-c1-0-6
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  + (1.41 + 2.44i)5-s + (−2 + 3.46i)11-s − 2.82·13-s + (−2.82 + 4.89i)17-s + (1.41 + 2.44i)19-s + (−1.49 + 2.59i)25-s − 2·29-s + (2.82 − 4.89i)31-s + (−5 − 8.66i)37-s − 5.65·41-s − 4·43-s + (2.82 + 4.89i)47-s + (3 − 5.19i)53-s − 11.3·55-s + (−1.41 + 2.44i)59-s + ⋯
L(s)  = 1  + (0.632 + 1.09i)5-s + (−0.603 + 1.04i)11-s − 0.784·13-s + (−0.685 + 1.18i)17-s + (0.324 + 0.561i)19-s + (−0.299 + 0.519i)25-s − 0.371·29-s + (0.508 − 0.879i)31-s + (−0.821 − 1.42i)37-s − 0.883·41-s − 0.609·43-s + (0.412 + 0.714i)47-s + (0.412 − 0.713i)53-s − 1.52·55-s + (−0.184 + 0.318i)59-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: $\chi_{3528} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3528,\ (\ :1/2),\ -0.991 + 0.126i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7786124470\)
\(L(\frac12)\) \(\approx\) \(0.7786124470\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.41 - 2.44i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (2 - 3.46i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 2.82T + 13T^{2} \)
17 \( 1 + (2.82 - 4.89i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.41 - 2.44i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 2T + 29T^{2} \)
31 \( 1 + (-2.82 + 4.89i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (5 + 8.66i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 5.65T + 41T^{2} \)
43 \( 1 + 4T + 43T^{2} \)
47 \( 1 + (-2.82 - 4.89i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3 + 5.19i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.41 - 2.44i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (7.07 + 12.2i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (6 - 10.3i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (4 + 6.92i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 14.1T + 83T^{2} \)
89 \( 1 + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 5.65T + 97T^{2} \)
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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

−9.087116562561012875635532133367, −8.037285382217116971362721765093, −7.42216467873141815767165790665, −6.73959917922075606231669446835, −6.06555108512956233430264424059, −5.25429733109148771289124931494, −4.35327058828926239790359437621, −3.40583360734403407060775456297, −2.35525569438083540615185774566, −1.87896230442102416641082244612, 0.21667541434842471700820729793, 1.34592546992437553205793238217, 2.52656453787305853083948607721, 3.29900396905184400265831926748, 4.75781796151735905809941936141, 4.98950292109763205398514837546, 5.76395253785010995859265351388, 6.72039512266687007379886006608, 7.43836703150968840344213292900, 8.432783435869701501411055883654

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