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Properties

Label 2-3528-504.59-c0-0-2
Degree $2$
Conductor $3528$
Sign $0.696 + 0.717i$
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 + (−0.382 − 0.923i)3-s + (0.499 + 0.866i)4-s + (−0.130 + 0.991i)6-s − 0.999i·8-s + (−0.707 + 0.707i)9-s + 0.517i·11-s + (0.608 − 0.793i)12-s + (−0.5 + 0.866i)16-s + (0.130 − 0.226i)17-s + (0.965 − 0.258i)18-s + (−1.05 + 0.608i)19-s + (0.258 − 0.448i)22-s + (−0.923 + 0.382i)24-s + 25-s + ⋯
L(s)  = 1  + (−0.866 − 0.5i)2-s + (−0.382 − 0.923i)3-s + (0.499 + 0.866i)4-s + (−0.130 + 0.991i)6-s − 0.999i·8-s + (−0.707 + 0.707i)9-s + 0.517i·11-s + (0.608 − 0.793i)12-s + (−0.5 + 0.866i)16-s + (0.130 − 0.226i)17-s + (0.965 − 0.258i)18-s + (−1.05 + 0.608i)19-s + (0.258 − 0.448i)22-s + (−0.923 + 0.382i)24-s + 25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.696 + 0.717i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.696 + 0.717i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3528\)    =    \(2^{3} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.696 + 0.717i$
Analytic conductor: \(1.76070\)
Root analytic conductor: \(1.32691\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3528} (2579, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3528,\ (\ :0),\ 0.696 + 0.717i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6721869528\)
\(L(\frac12)\) \(\approx\) \(0.6721869528\)
\(L(1)\) 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 \( 1 + (0.382 + 0.923i)T \)
7 \( 1 \)
good5 \( 1 - T^{2} \)
11 \( 1 - 0.517iT - T^{2} \)
13 \( 1 + (-0.5 - 0.866i)T^{2} \)
17 \( 1 + (-0.130 + 0.226i)T + (-0.5 - 0.866i)T^{2} \)
19 \( 1 + (1.05 - 0.608i)T + (0.5 - 0.866i)T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 + (-0.5 + 0.866i)T^{2} \)
31 \( 1 + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + (0.5 - 0.866i)T^{2} \)
41 \( 1 + (-0.991 + 1.71i)T + (-0.5 - 0.866i)T^{2} \)
43 \( 1 + (-0.965 - 1.67i)T + (-0.5 + 0.866i)T^{2} \)
47 \( 1 + (0.5 + 0.866i)T^{2} \)
53 \( 1 + (-0.5 - 0.866i)T^{2} \)
59 \( 1 + (0.793 + 1.37i)T + (-0.5 + 0.866i)T^{2} \)
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 \( 1 + (-1.37 - 0.793i)T + (0.5 + 0.866i)T^{2} \)
79 \( 1 + (0.5 + 0.866i)T^{2} \)
83 \( 1 + (-0.923 - 1.60i)T + (-0.5 + 0.866i)T^{2} \)
89 \( 1 + (-0.382 - 0.662i)T + (-0.5 + 0.866i)T^{2} \)
97 \( 1 + (-1.71 + 0.991i)T + (0.5 - 0.866i)T^{2} \)
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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

−8.571860536441435720503962749584, −7.954759923644099012557573990978, −7.28604509953972472383491995014, −6.64684237384036550414239702421, −5.94384820807946147274887538469, −4.84592449073699622343235367249, −3.83011876124829947147156682526, −2.66735393343724297261621220472, −1.99196252910902377694082116764, −0.901143901711302470200945087408, 0.75490932382727702608727315719, 2.31265638023995680367243360431, 3.33959257166698745708405786461, 4.48548126890727522324084869170, 5.11496297983157948944836432772, 6.08807799339822934423484238802, 6.42714106976649663549093580958, 7.45555154669993713340070873812, 8.275010000593919438391251137935, 9.024013830343133976192402760497

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