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Properties

Label 2-3528-168.53-c0-0-3
Degree $2$
Conductor $3528$
Sign $0.956 - 0.292i$
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  + (0.965 + 0.258i)2-s + (0.866 + 0.499i)4-s + (0.707 + 0.707i)8-s + (0.707 − 1.22i)11-s + (0.500 + 0.866i)16-s + (1 − 0.999i)22-s + (1.22 − 0.707i)23-s + (0.5 − 0.866i)25-s − 1.41·29-s + (0.258 + 0.965i)32-s + (−1.73 + i)37-s + 2i·43-s + (1.22 − 0.707i)44-s + (1.36 − 0.366i)46-s + (0.707 − 0.707i)50-s + ⋯
L(s)  = 1  + (0.965 + 0.258i)2-s + (0.866 + 0.499i)4-s + (0.707 + 0.707i)8-s + (0.707 − 1.22i)11-s + (0.500 + 0.866i)16-s + (1 − 0.999i)22-s + (1.22 − 0.707i)23-s + (0.5 − 0.866i)25-s − 1.41·29-s + (0.258 + 0.965i)32-s + (−1.73 + i)37-s + 2i·43-s + (1.22 − 0.707i)44-s + (1.36 − 0.366i)46-s + (0.707 − 0.707i)50-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.956 - 0.292i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.956 - 0.292i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3528\)    =    \(2^{3} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.956 - 0.292i$
Analytic conductor: \(1.76070\)
Root analytic conductor: \(1.32691\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3528} (557, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3528,\ (\ :0),\ 0.956 - 0.292i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.524367581\)
\(L(\frac12)\) \(\approx\) \(2.524367581\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.965 - 0.258i)T \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (-0.707 + 1.22i)T + (-0.5 - 0.866i)T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (-1.22 + 0.707i)T + (0.5 - 0.866i)T^{2} \)
29 \( 1 + 1.41T + T^{2} \)
31 \( 1 + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + (1.73 - i)T + (0.5 - 0.866i)T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - 2iT - T^{2} \)
47 \( 1 + (0.5 - 0.866i)T^{2} \)
53 \( 1 + (-0.707 + 1.22i)T + (-0.5 - 0.866i)T^{2} \)
59 \( 1 + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T^{2} \)
71 \( 1 - 1.41iT - T^{2} \)
73 \( 1 + (-0.5 - 0.866i)T^{2} \)
79 \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + (0.5 - 0.866i)T^{2} \)
97 \( 1 + T^{2} \)
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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

−8.525253073064776254696485209032, −8.109992073056314108110650639679, −6.89849937823989381612783150564, −6.63600041577702225261930857513, −5.70399735175413633775649199125, −5.06125120479342708265635400790, −4.14825747851283387746418981018, −3.37911442612051817460154272850, −2.64286750272634784815737227313, −1.32630701332569949923969441657, 1.45218354717056801190171668691, 2.22089162242465677920626179197, 3.44517019029552276025661933458, 3.95155806147500941669344675898, 5.02512482830077618203902916700, 5.41425685842757303311331204694, 6.47666230277594138580857413068, 7.24856678197594817654074170393, 7.47474952149109739337638679741, 9.133296599590962365946464763225

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