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Properties

Label 2-3528-72.67-c0-0-7
Degree $2$
Conductor $3528$
Sign $-0.766 + 0.642i$
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  + (−0.5 − 0.866i)2-s + (0.866 − 0.5i)3-s + (−0.499 + 0.866i)4-s + (−0.866 − 0.499i)6-s + 0.999·8-s + (0.499 − 0.866i)9-s + (−0.5 − 0.866i)11-s + 0.999i·12-s + (−0.5 − 0.866i)16-s − 1.73·17-s − 0.999·18-s + 1.73·19-s + (−0.499 + 0.866i)22-s + (0.866 − 0.5i)24-s + (−0.5 − 0.866i)25-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)2-s + (0.866 − 0.5i)3-s + (−0.499 + 0.866i)4-s + (−0.866 − 0.499i)6-s + 0.999·8-s + (0.499 − 0.866i)9-s + (−0.5 − 0.866i)11-s + 0.999i·12-s + (−0.5 − 0.866i)16-s − 1.73·17-s − 0.999·18-s + 1.73·19-s + (−0.499 + 0.866i)22-s + (0.866 − 0.5i)24-s + (−0.5 − 0.866i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3528\)    =    \(2^{3} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.766 + 0.642i$
Analytic conductor: \(1.76070\)
Root analytic conductor: \(1.32691\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3528} (3235, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3528,\ (\ :0),\ -0.766 + 0.642i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.128393517\)
\(L(\frac12)\) \(\approx\) \(1.128393517\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 + 0.866i)T \)
3 \( 1 + (-0.866 + 0.5i)T \)
7 \( 1 \)
good5 \( 1 + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + (0.5 + 0.866i)T^{2} \)
17 \( 1 + 1.73T + T^{2} \)
19 \( 1 - 1.73T + T^{2} \)
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 \( 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2} \)
43 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
47 \( 1 + (0.5 - 0.866i)T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - 1.73T + T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (-0.5 + 0.866i)T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \)
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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

−8.626548594513268137497645627391, −7.88112191686739295363481958541, −7.31416730295540408200044812637, −6.45904560032753255785622720586, −5.33669425285980143677283930936, −4.25714491406107123714254798086, −3.50462363960872984512341434575, −2.68603060676965850021642576473, −1.99808935674086793156071730533, −0.70994332287186614771724499622, 1.56910118947029171975943124022, 2.56839747743186034436499420823, 3.71835541975936937054137983387, 4.73035543634894540203255999174, 5.05920633259048110981569120012, 6.20208793300539520749907447819, 7.08642865150607511677901833763, 7.64059447093327391591002648970, 8.211623799069015982041501846470, 9.085100746353997677037460231511

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