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Properties

Label 2-3528-72.67-c0-0-1
Degree $2$
Conductor $3528$
Sign $-0.642 - 0.766i$
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 + (−0.965 − 0.258i)3-s + (−0.499 + 0.866i)4-s + (−0.258 − 0.965i)6-s − 0.999·8-s + (0.866 + 0.499i)9-s + (0.866 + 1.5i)11-s + (0.707 − 0.707i)12-s + (−0.5 − 0.866i)16-s − 0.517·17-s + i·18-s + 1.93·19-s + (−0.866 + 1.5i)22-s + (0.965 + 0.258i)24-s + (−0.5 − 0.866i)25-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)2-s + (−0.965 − 0.258i)3-s + (−0.499 + 0.866i)4-s + (−0.258 − 0.965i)6-s − 0.999·8-s + (0.866 + 0.499i)9-s + (0.866 + 1.5i)11-s + (0.707 − 0.707i)12-s + (−0.5 − 0.866i)16-s − 0.517·17-s + i·18-s + 1.93·19-s + (−0.866 + 1.5i)22-s + (0.965 + 0.258i)24-s + (−0.5 − 0.866i)25-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(3528\)    =    \(2^{3} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.642 - 0.766i$
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: \((2,\ 3528,\ (\ :0),\ -0.642 - 0.766i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.096877164\)
\(L(\frac12)\) \(\approx\) \(1.096877164\)
\(L(1)\) 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 \( 1 + (0.965 + 0.258i)T \)
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 \( 1 + 0.517T + T^{2} \)
19 \( 1 - 1.93T + T^{2} \)
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 \( 1 + (0.258 - 0.448i)T + (-0.5 - 0.866i)T^{2} \)
43 \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \)
47 \( 1 + (0.5 - 0.866i)T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (0.965 - 1.67i)T + (-0.5 - 0.866i)T^{2} \)
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 \( 1 + 1.93T + T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (-0.707 - 1.22i)T + (-0.5 + 0.866i)T^{2} \)
89 \( 1 - 1.41T + T^{2} \)
97 \( 1 + (-0.258 - 0.448i)T + (-0.5 + 0.866i)T^{2} \)
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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

−9.053416904386050352358941257743, −7.78904208014900166401549311105, −7.43501447351926826291951268661, −6.70069649241095592425434472755, −6.11698779959248346345350060105, −5.30336960586070234839810327898, −4.55376371257396027600576767475, −4.05966781075213181939715056127, −2.72310044057146837383441963864, −1.35878513977873447845626586687, 0.69972614601729358304220139664, 1.69295068886782093986890929912, 3.24563105715642110575072738644, 3.66489657535822245875869029983, 4.65387500742361836807957038619, 5.48516395375779874118669280769, 5.90703758771475843000734531615, 6.71090479080190401887915305466, 7.64887656819359523692325517877, 8.968054224238630657472234403754

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