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Properties

Label 2-3528-504.419-c0-0-1
Degree $2$
Conductor $3528$
Sign $0.944 - 0.328i$
Analytic cond. $1.76070$
Root an. cond. $1.32691$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.866 + 0.5i)2-s + (−0.991 + 0.130i)3-s + (0.499 − 0.866i)4-s + (0.793 − 0.608i)6-s + 0.999i·8-s + (0.965 − 0.258i)9-s + (1.67 − 0.965i)11-s + (−0.382 + 0.923i)12-s + (−0.5 − 0.866i)16-s + 1.58·17-s + (−0.707 + 0.707i)18-s + 1.98i·19-s + (−0.965 + 1.67i)22-s + (−0.130 − 0.991i)24-s + (−0.5 − 0.866i)25-s + ⋯
L(s)  = 1  + (−0.866 + 0.5i)2-s + (−0.991 + 0.130i)3-s + (0.499 − 0.866i)4-s + (0.793 − 0.608i)6-s + 0.999i·8-s + (0.965 − 0.258i)9-s + (1.67 − 0.965i)11-s + (−0.382 + 0.923i)12-s + (−0.5 − 0.866i)16-s + 1.58·17-s + (−0.707 + 0.707i)18-s + 1.98i·19-s + (−0.965 + 1.67i)22-s + (−0.130 − 0.991i)24-s + (−0.5 − 0.866i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.944 - 0.328i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.944 - 0.328i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3528\)    =    \(2^{3} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.944 - 0.328i$
Analytic conductor: \(1.76070\)
Root analytic conductor: \(1.32691\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3528} (2939, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3528,\ (\ :0),\ 0.944 - 0.328i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7227346344\)
\(L(\frac12)\) \(\approx\) \(0.7227346344\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 - 0.5i)T \)
3 \( 1 + (0.991 - 0.130i)T \)
7 \( 1 \)
good5 \( 1 + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (-1.67 + 0.965i)T + (0.5 - 0.866i)T^{2} \)
13 \( 1 + (-0.5 - 0.866i)T^{2} \)
17 \( 1 - 1.58T + T^{2} \)
19 \( 1 - 1.98iT - T^{2} \)
23 \( 1 + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + (-0.5 + 0.866i)T^{2} \)
31 \( 1 + (-0.5 - 0.866i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (-0.608 + 1.05i)T + (-0.5 - 0.866i)T^{2} \)
43 \( 1 + (0.258 + 0.448i)T + (-0.5 + 0.866i)T^{2} \)
47 \( 1 + (0.5 - 0.866i)T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + (-0.130 + 0.226i)T + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + 0.261iT - T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (0.923 + 1.60i)T + (-0.5 + 0.866i)T^{2} \)
89 \( 1 - 0.765T + T^{2} \)
97 \( 1 + (1.05 - 0.608i)T + (0.5 - 0.866i)T^{2} \)
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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

−8.786273071211428649277358440594, −8.013001791637061061919000836854, −7.32272057604350843784106952932, −6.43401142154095805770639256871, −5.89134811085018574899562205830, −5.51331678658848956383359581460, −4.20115578569037242415082448690, −3.46441833568446863069153926774, −1.71001700211371802139671784386, −0.929289765004691417170128259017, 0.999743004242491795538438858387, 1.75835996569407429530150658004, 3.07065720936869926172850120887, 4.08135233888245856816622085931, 4.79806723620618234795537303066, 5.89808513355176895056135220629, 6.74193442027231730159091740915, 7.17316327679856959471396488310, 7.86930060240520849958467765358, 8.970479745907726558574630145658

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