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Properties

Label 2-3528-21.17-c1-0-15
Degree $2$
Conductor $3528$
Sign $0.999 - 0.0285i$
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  + (−0.765 − 1.32i)5-s + (0.507 + 0.292i)11-s − 2.16i·13-s + (−2.93 + 5.07i)17-s + (−4.52 + 2.61i)19-s + (1.94 − 1.12i)23-s + (1.32 − 2.30i)25-s + 5.41i·29-s + (3.74 + 2.16i)31-s + (−2 − 3.46i)37-s + 8.92·41-s + 10.4·43-s + (3.69 + 6.40i)47-s + (4.68 + 2.70i)53-s − 0.896i·55-s + ⋯
L(s)  = 1  + (−0.342 − 0.592i)5-s + (0.152 + 0.0883i)11-s − 0.600i·13-s + (−0.710 + 1.23i)17-s + (−1.03 + 0.599i)19-s + (0.404 − 0.233i)23-s + (0.265 − 0.460i)25-s + 1.00i·29-s + (0.673 + 0.388i)31-s + (−0.328 − 0.569i)37-s + 1.39·41-s + 1.59·43-s + (0.539 + 0.933i)47-s + (0.644 + 0.371i)53-s − 0.120i·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0285i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0285i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3528\)    =    \(2^{3} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.999 - 0.0285i$
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: \((2,\ 3528,\ (\ :1/2),\ 0.999 - 0.0285i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.564887717\)
\(L(\frac12)\) \(\approx\) \(1.564887717\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.765 + 1.32i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-0.507 - 0.292i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 2.16iT - 13T^{2} \)
17 \( 1 + (2.93 - 5.07i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (4.52 - 2.61i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.94 + 1.12i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 5.41iT - 29T^{2} \)
31 \( 1 + (-3.74 - 2.16i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (2 + 3.46i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 8.92T + 41T^{2} \)
43 \( 1 - 10.4T + 43T^{2} \)
47 \( 1 + (-3.69 - 6.40i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-4.68 - 2.70i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.82 + 8.36i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 4.58iT - 71T^{2} \)
73 \( 1 + (10.9 + 6.30i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-1.17 - 2.02i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 13.5T + 83T^{2} \)
89 \( 1 + (2.93 + 5.07i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 8.28iT - 97T^{2} \)
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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

−8.690768431138227589834789570806, −7.938955194212058050919433936178, −7.15852340798663026244254763812, −6.22627469146994915689950973404, −5.68030427170139644371251266994, −4.49848136246525386075425316109, −4.17936984858789552716750625930, −3.02968508501070288807188016052, −1.96486103231486648865563476375, −0.797096939138110808563231183858, 0.67136743117451584055438452296, 2.24090315776931309681842489388, 2.86643434829944427319199618963, 4.04271050845528872003780075058, 4.56287485664164636110599776756, 5.61325077130236601268201033824, 6.50819077976241811122376892133, 7.06758188956308643652077610736, 7.66322736136476802349892332129, 8.733333968179003357451223579377

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