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Properties

Label 2-3528-7.4-c1-0-45
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 + (−1 + 1.73i)11-s + 5.65·13-s + (−1.41 + 2.44i)17-s + (−2.82 − 4.89i)19-s + (−3 − 5.19i)23-s + (−1.49 + 2.59i)25-s − 4·29-s + (2.82 − 4.89i)31-s + (1 + 1.73i)37-s − 2.82·41-s − 4·43-s + (5.65 + 9.79i)47-s + (−6 + 10.3i)53-s + 5.65·55-s + ⋯
L(s)  = 1  + (−0.632 − 1.09i)5-s + (−0.301 + 0.522i)11-s + 1.56·13-s + (−0.342 + 0.594i)17-s + (−0.648 − 1.12i)19-s + (−0.625 − 1.08i)23-s + (−0.299 + 0.519i)25-s − 0.742·29-s + (0.508 − 0.879i)31-s + (0.164 + 0.284i)37-s − 0.441·41-s − 0.609·43-s + (0.825 + 1.42i)47-s + (−0.824 + 1.42i)53-s + 0.762·55-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

\(L(1)\) \(\approx\) \(0.6279828940\)
\(L(\frac12)\) \(\approx\) \(0.6279828940\)
\(L(\frac{3}{2})\) 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 \( 1 + (1 - 1.73i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 5.65T + 13T^{2} \)
17 \( 1 + (1.41 - 2.44i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.82 + 4.89i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (3 + 5.19i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 4T + 29T^{2} \)
31 \( 1 + (-2.82 + 4.89i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1 - 1.73i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 2.82T + 41T^{2} \)
43 \( 1 + 4T + 43T^{2} \)
47 \( 1 + (-5.65 - 9.79i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (6 - 10.3i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-5.65 + 9.79i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.82 + 4.89i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6 + 10.3i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 6T + 71T^{2} \)
73 \( 1 + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (4 + 6.92i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 11.3T + 83T^{2} \)
89 \( 1 + (4.24 + 7.34i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 11.3T + 97T^{2} \)
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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

−8.201868430388122185112348904984, −7.78333486263716821958233487865, −6.55950255130791717461790073068, −6.10206686275671847625346843454, −4.95553250835819972330296838879, −4.38064270842646198907727175390, −3.74858046341464022964079554106, −2.47274397015080130763037242510, −1.35756320633370239334647848808, −0.19499898755856863083878066500, 1.44915848150364844547056413159, 2.65699591385508786614902251628, 3.68088204535689104773267375777, 3.85111824381347163125414451099, 5.31958347556492921911805582203, 5.99975163820132977448402643375, 6.75582501516725827036933261681, 7.38746469539236020439903240139, 8.284128055691339425415691529969, 8.630803386858349803889810384604

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