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Properties

Label 2-3528-168.59-c0-0-0
Degree $2$
Conductor $3528$
Sign $0.726 + 0.686i$
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  + (0.258 − 0.965i)2-s + (−0.866 − 0.499i)4-s + (−0.707 + 0.707i)8-s + (1.22 + 0.707i)11-s + (0.500 + 0.866i)16-s + (1 − 0.999i)22-s + (0.707 + 1.22i)23-s + (−0.5 + 0.866i)25-s + 1.41·29-s + (0.965 − 0.258i)32-s + (−1.73 + i)37-s + (−0.707 − 1.22i)44-s + (1.36 − 0.366i)46-s + (0.707 + 0.707i)50-s + (0.707 − 1.22i)53-s + ⋯
L(s)  = 1  + (0.258 − 0.965i)2-s + (−0.866 − 0.499i)4-s + (−0.707 + 0.707i)8-s + (1.22 + 0.707i)11-s + (0.500 + 0.866i)16-s + (1 − 0.999i)22-s + (0.707 + 1.22i)23-s + (−0.5 + 0.866i)25-s + 1.41·29-s + (0.965 − 0.258i)32-s + (−1.73 + i)37-s + (−0.707 − 1.22i)44-s + (1.36 − 0.366i)46-s + (0.707 + 0.707i)50-s + (0.707 − 1.22i)53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.726 + 0.686i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.726 + 0.686i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3528\)    =    \(2^{3} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.726 + 0.686i$
Analytic conductor: \(1.76070\)
Root analytic conductor: \(1.32691\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3528} (1403, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3528,\ (\ :0),\ 0.726 + 0.686i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.380222521\)
\(L(\frac12)\) \(\approx\) \(1.380222521\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.258 + 0.965i)T \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (-1.22 - 0.707i)T + (0.5 + 0.866i)T^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 + (-0.5 - 0.866i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.707 - 1.22i)T + (-0.5 + 0.866i)T^{2} \)
29 \( 1 - 1.41T + T^{2} \)
31 \( 1 + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + (1.73 - i)T + (0.5 - 0.866i)T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + (0.5 - 0.866i)T^{2} \)
53 \( 1 + (-0.707 + 1.22i)T + (-0.5 - 0.866i)T^{2} \)
59 \( 1 + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 - 1.41T + T^{2} \)
73 \( 1 + (0.5 + 0.866i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + (-0.5 + 0.866i)T^{2} \)
97 \( 1 - T^{2} \)
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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

−8.884555574498939132497389245930, −8.136196113736627041831518184045, −7.05569232645121408037593434522, −6.43562764158603928984961105537, −5.34700900326431833109729184901, −4.79673297294158031673034203932, −3.79875582812275682665550535434, −3.26899926929455470983045365633, −2.01885182563312885816428248709, −1.23844819419050847025256862902, 0.904979083194529587460774075689, 2.58966208542478317743268166827, 3.64685249252020382249505325622, 4.28117733334761202681365142535, 5.14116651855249354344358343557, 5.99713625458241546657323341157, 6.62475416398687705227348808946, 7.12434524397168184543890269904, 8.225596442969306602008673647371, 8.650139706603085326025513958949

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