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Properties

Label 2-3528-7.4-c1-0-49
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  + (−1.63 − 2.83i)5-s + (1.63 − 2.83i)11-s − 6.27·13-s + (2 − 3.46i)17-s + (−3.13 − 5.43i)19-s + (2 + 3.46i)23-s + (−2.86 + 4.95i)25-s − 5.27·29-s + (−0.5 + 0.866i)31-s + (1.13 + 1.97i)37-s − 4.54·41-s + 0.274·43-s + (3 + 5.19i)47-s + (4.63 − 8.03i)53-s − 10.7·55-s + ⋯
L(s)  = 1  + (−0.732 − 1.26i)5-s + (0.493 − 0.855i)11-s − 1.74·13-s + (0.485 − 0.840i)17-s + (−0.719 − 1.24i)19-s + (0.417 + 0.722i)23-s + (−0.572 + 0.991i)25-s − 0.979·29-s + (−0.0898 + 0.155i)31-s + (0.186 + 0.323i)37-s − 0.710·41-s + 0.0419·43-s + (0.437 + 0.757i)47-s + (0.637 − 1.10i)53-s − 1.44·55-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

\(L(1)\) \(\approx\) \(0.2689407533\)
\(L(\frac12)\) \(\approx\) \(0.2689407533\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.63 + 2.83i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.63 + 2.83i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 6.27T + 13T^{2} \)
17 \( 1 + (-2 + 3.46i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.13 + 5.43i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2 - 3.46i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 5.27T + 29T^{2} \)
31 \( 1 + (0.5 - 0.866i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.13 - 1.97i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 4.54T + 41T^{2} \)
43 \( 1 - 0.274T + 43T^{2} \)
47 \( 1 + (-3 - 5.19i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-4.63 + 8.03i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-0.637 + 1.10i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5 - 8.66i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.137 + 0.238i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 2T + 71T^{2} \)
73 \( 1 + (2.13 - 3.70i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (5.77 + 10.0i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 7.27T + 83T^{2} \)
89 \( 1 + (-5.27 - 9.13i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 8.72T + 97T^{2} \)
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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

−8.108645904929521930204200617885, −7.41222525439851309273238491294, −6.80565481905194251580174009318, −5.55680545253185996495740135826, −5.00314489569266071102728237433, −4.37311254869851507936916089953, −3.41644564917240191441888541885, −2.40603133068610929621282892548, −1.01992446524245547949977682468, −0.090562831323365481402228476816, 1.85786668033788830292703780484, 2.65307693125030121054280548218, 3.69721416794052783552388272573, 4.23725055952353828678947301846, 5.27206336187944315418307473471, 6.25248078262406931852750068798, 6.97902166295105353337455490434, 7.47102052439004550852374000007, 8.071635381845116035518933480966, 9.066222415096811429015049433646

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