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Properties

Label 2-3528-21.17-c1-0-7
Degree $2$
Conductor $3528$
Sign $-0.867 - 0.496i$
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  + (0.923 + 1.60i)5-s + (1.73 + i)11-s + 5.54i·13-s + (−2.23 + 3.86i)17-s + (−1.32 + 0.765i)19-s + (−4.18 + 2.41i)23-s + (0.792 − 1.37i)25-s + 0.828i·29-s + (−1.32 − 0.765i)31-s + (−0.707 − 1.22i)37-s + 0.765·41-s − 10.8·43-s + (−4.46 − 7.72i)47-s + (−0.210 − 0.121i)53-s + 3.69i·55-s + ⋯
L(s)  = 1  + (0.413 + 0.715i)5-s + (0.522 + 0.301i)11-s + 1.53i·13-s + (−0.540 + 0.936i)17-s + (−0.304 + 0.175i)19-s + (−0.871 + 0.503i)23-s + (0.158 − 0.274i)25-s + 0.153i·29-s + (−0.238 − 0.137i)31-s + (−0.116 − 0.201i)37-s + 0.119·41-s − 1.65·43-s + (−0.650 − 1.12i)47-s + (−0.0288 − 0.0166i)53-s + 0.498i·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.867 - 0.496i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.867 - 0.496i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3528\)    =    \(2^{3} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.867 - 0.496i$
Analytic conductor: \(28.1712\)
Root analytic conductor: \(5.30765\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3528} (521, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3528,\ (\ :1/2),\ -0.867 - 0.496i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.236716907\)
\(L(\frac12)\) \(\approx\) \(1.236716907\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.923 - 1.60i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.73 - i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 5.54iT - 13T^{2} \)
17 \( 1 + (2.23 - 3.86i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.32 - 0.765i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (4.18 - 2.41i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 0.828iT - 29T^{2} \)
31 \( 1 + (1.32 + 0.765i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.707 + 1.22i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 0.765T + 41T^{2} \)
43 \( 1 + 10.8T + 43T^{2} \)
47 \( 1 + (4.46 + 7.72i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.210 + 0.121i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.39 - 2.42i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.21 + 0.699i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-7.41 + 12.8i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 15.6iT - 71T^{2} \)
73 \( 1 + (7.84 + 4.52i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (7.65 + 13.2i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 13.5T + 83T^{2} \)
89 \( 1 + (-4.93 - 8.55i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 16.1iT - 97T^{2} \)
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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

−8.903598376204174604437872530751, −8.213567901689614109914188587074, −7.22549174669029823940308878968, −6.49996410219906476071457699122, −6.24630763186913890998838222013, −5.05825004805226404233380221890, −4.14929322188071611938773984729, −3.53114385029261325854807602572, −2.19492225246339408546887084741, −1.69660086697510319194752077346, 0.34993775604077913044155352832, 1.45598028537839043050867619476, 2.62848966950346826030762445142, 3.49324340783546977043979636133, 4.57933304656417263128508332237, 5.20486999728783945175058276086, 5.95640843901391126273473955405, 6.70140847682265764266522650363, 7.62095933209247901644613014760, 8.387667273475847680362783551488

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