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Properties

Label 2-3528-504.499-c0-0-3
Degree $2$
Conductor $3528$
Sign $0.101 + 0.994i$
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  − 2-s + (−0.965 + 0.258i)3-s + 4-s + (0.965 − 0.258i)6-s − 8-s + (0.866 − 0.499i)9-s + (−0.866 + 1.5i)11-s + (−0.965 + 0.258i)12-s + 16-s + (−0.965 − 1.67i)17-s + (−0.866 + 0.499i)18-s + (0.258 − 0.448i)19-s + (0.866 − 1.5i)22-s + (0.965 − 0.258i)24-s + (−0.5 + 0.866i)25-s + ⋯
L(s)  = 1  − 2-s + (−0.965 + 0.258i)3-s + 4-s + (0.965 − 0.258i)6-s − 8-s + (0.866 − 0.499i)9-s + (−0.866 + 1.5i)11-s + (−0.965 + 0.258i)12-s + 16-s + (−0.965 − 1.67i)17-s + (−0.866 + 0.499i)18-s + (0.258 − 0.448i)19-s + (0.866 − 1.5i)22-s + (0.965 − 0.258i)24-s + (−0.5 + 0.866i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.101 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.101 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3528\)    =    \(2^{3} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.101 + 0.994i$
Analytic conductor: \(1.76070\)
Root analytic conductor: \(1.32691\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3528} (3019, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3528,\ (\ :0),\ 0.101 + 0.994i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2997471625\)
\(L(\frac12)\) \(\approx\) \(0.2997471625\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 + (0.965 - 0.258i)T \)
7 \( 1 \)
good5 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \)
13 \( 1 + (0.5 + 0.866i)T^{2} \)
17 \( 1 + (0.965 + 1.67i)T + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + (-0.258 + 0.448i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.5 - 0.866i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + (0.5 + 0.866i)T^{2} \)
41 \( 1 + (-0.965 + 1.67i)T + (-0.5 - 0.866i)T^{2} \)
43 \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + (0.5 - 0.866i)T^{2} \)
59 \( 1 + 0.517T + T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 - T + T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (0.258 + 0.448i)T + (-0.5 + 0.866i)T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + (-0.707 - 1.22i)T + (-0.5 + 0.866i)T^{2} \)
89 \( 1 + (0.707 - 1.22i)T + (-0.5 - 0.866i)T^{2} \)
97 \( 1 + (0.965 + 1.67i)T + (-0.5 + 0.866i)T^{2} \)
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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

−8.775518945383998223027681665398, −7.56370415023999825833655228447, −7.18085343124752415305850729779, −6.66946885238779213561910208126, −5.46615362898364207597821799874, −5.05263843800033035477368139935, −4.01310110479768577586428249431, −2.68313586611384699119992140909, −1.83032188867218562234625004807, −0.30804634057613990394717482162, 1.09187349884919800982884763820, 2.16267688771631391679190881224, 3.25819151814740815440111638979, 4.38628518329267140993718075014, 5.51900879188481436689128216461, 6.24211251825073740601903506403, 6.43149218168277156729867548619, 7.73230111762215720513708064794, 8.085788797599974932307357964648, 8.747676552168635398248950295716

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