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Properties

Label 2-3528-7.2-c1-0-34
Degree $2$
Conductor $3528$
Sign $-0.266 + 0.963i$
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 − 6·13-s + (1 + 1.73i)17-s + (2 − 3.46i)19-s + (−2 + 3.46i)23-s + (0.500 + 0.866i)25-s + 10·29-s + (−4 − 6.92i)31-s + (−3 + 5.19i)37-s − 2·41-s − 4·43-s + (−4 + 6.92i)47-s + (−5 − 8.66i)53-s + (−6 − 10.3i)59-s + (−1 + 1.73i)61-s + ⋯
L(s)  = 1  + (−0.447 + 0.774i)5-s − 1.66·13-s + (0.242 + 0.420i)17-s + (0.458 − 0.794i)19-s + (−0.417 + 0.722i)23-s + (0.100 + 0.173i)25-s + 1.85·29-s + (−0.718 − 1.24i)31-s + (−0.493 + 0.854i)37-s − 0.312·41-s − 0.609·43-s + (−0.583 + 1.01i)47-s + (−0.686 − 1.18i)53-s + (−0.781 − 1.35i)59-s + (−0.128 + 0.221i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.266 + 0.963i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.266 + 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3528\)    =    \(2^{3} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.266 + 0.963i$
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: \((2,\ 3528,\ (\ :1/2),\ -0.266 + 0.963i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5065948624\)
\(L(\frac12)\) \(\approx\) \(0.5065948624\)
\(L(\frac{3}{2})\) 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 \( 1 + 6T + 13T^{2} \)
17 \( 1 + (-1 - 1.73i)T + (-8.5 + 14.7i)T^{2} \)
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 \( 1 - 10T + 29T^{2} \)
31 \( 1 + (4 + 6.92i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (3 - 5.19i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 2T + 41T^{2} \)
43 \( 1 + 4T + 43T^{2} \)
47 \( 1 + (4 - 6.92i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (5 + 8.66i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (6 + 10.3i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1 - 1.73i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (6 + 10.3i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 12T + 71T^{2} \)
73 \( 1 + (7 + 12.1i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4 + 6.92i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 12T + 83T^{2} \)
89 \( 1 + (-1 + 1.73i)T + (-44.5 - 77.0i)T^{2} \)
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

−8.039157493234952738651435269704, −7.71084010956113871197564615937, −6.86697632945751690128319984656, −6.33412977954345192539242047627, −5.13972949171431704581934301483, −4.65586883738171795913397678717, −3.44944934689807826946092778836, −2.88148956732844158701413511855, −1.81349383031692935641239444720, −0.16400404974601194804692622465, 1.08633607958420632713005377265, 2.35005770868575698380149320852, 3.26544837931114062552187486371, 4.37328711784558422094157111598, 4.90874190317743168797615358570, 5.59848273141288266590724538708, 6.73219651372610855731367190477, 7.31666385090932706879795966349, 8.154117198051418666831477395359, 8.640999272341286951049326553436

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