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Properties

Label 2-3528-504.419-c0-0-0
Degree $2$
Conductor $3528$
Sign $0.242 - 0.970i$
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 + (−0.130 − 0.991i)3-s + (0.499 − 0.866i)4-s + (0.608 + 0.793i)6-s + 0.999i·8-s + (−0.965 + 0.258i)9-s + (−1.67 + 0.965i)11-s + (−0.923 − 0.382i)12-s + (−0.5 − 0.866i)16-s + 1.21·17-s + (0.707 − 0.707i)18-s + 0.261i·19-s + (0.965 − 1.67i)22-s + (0.991 − 0.130i)24-s + (−0.5 − 0.866i)25-s + ⋯
L(s)  = 1  + (−0.866 + 0.5i)2-s + (−0.130 − 0.991i)3-s + (0.499 − 0.866i)4-s + (0.608 + 0.793i)6-s + 0.999i·8-s + (−0.965 + 0.258i)9-s + (−1.67 + 0.965i)11-s + (−0.923 − 0.382i)12-s + (−0.5 − 0.866i)16-s + 1.21·17-s + (0.707 − 0.707i)18-s + 0.261i·19-s + (0.965 − 1.67i)22-s + (0.991 − 0.130i)24-s + (−0.5 − 0.866i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.242 - 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.242 - 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3528\)    =    \(2^{3} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.242 - 0.970i$
Analytic conductor: \(1.76070\)
Root analytic conductor: \(1.32691\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3528} (2939, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3528,\ (\ :0),\ 0.242 - 0.970i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4618397771\)
\(L(\frac12)\) \(\approx\) \(0.4618397771\)
\(L(1)\) 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 \( 1 + (0.130 + 0.991i)T \)
7 \( 1 \)
good5 \( 1 + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (1.67 - 0.965i)T + (0.5 - 0.866i)T^{2} \)
13 \( 1 + (-0.5 - 0.866i)T^{2} \)
17 \( 1 - 1.21T + T^{2} \)
19 \( 1 - 0.261iT - T^{2} \)
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 \( 1 + (0.793 - 1.37i)T + (-0.5 - 0.866i)T^{2} \)
43 \( 1 + (-0.258 - 0.448i)T + (-0.5 + 0.866i)T^{2} \)
47 \( 1 + (0.5 - 0.866i)T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + (0.991 - 1.71i)T + (-0.5 - 0.866i)T^{2} \)
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 \( 1 - 1.98iT - T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (-0.382 - 0.662i)T + (-0.5 + 0.866i)T^{2} \)
89 \( 1 - 1.84T + T^{2} \)
97 \( 1 + (-1.37 + 0.793i)T + (0.5 - 0.866i)T^{2} \)
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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

−8.600187651175577607066687964574, −7.923301108870627126293792712334, −7.62942233497611769174648030509, −6.91626117182716359791313353627, −6.01442136393881073589190649356, −5.47169111354512640093591025103, −4.65963642619755271153690374794, −2.94463739807692680315814953332, −2.23488287064216846161298092928, −1.19120073182921632226156055535, 0.38602285949617989978821013844, 2.08440939855505136947756484737, 3.26918123330825873124953318491, 3.42365762332981742516704063107, 4.81371631934279557705185845304, 5.50787895003418424006173432130, 6.27705966406622724941478310659, 7.58000151167132674261616882032, 7.920985268237436677493130053315, 8.778178420353285609838963442110

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