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Properties

Label 2-3528-7.4-c1-0-41
Degree $2$
Conductor $3528$
Sign $-0.991 + 0.126i$
Analytic cond. $28.1712$
Root an. cond. $5.30765$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1 − 1.73i)5-s + (−1 + 1.73i)11-s + 2·13-s + (−3 + 5.19i)17-s + (2 + 3.46i)19-s + (−3 − 5.19i)23-s + (0.500 − 0.866i)25-s + (2 − 3.46i)31-s + (−5 − 8.66i)37-s + 2·41-s − 4·43-s + (−2 − 3.46i)47-s + (6 − 10.3i)53-s + 3.99·55-s + (−6 + 10.3i)59-s + ⋯
L(s)  = 1  + (−0.447 − 0.774i)5-s + (−0.301 + 0.522i)11-s + 0.554·13-s + (−0.727 + 1.26i)17-s + (0.458 + 0.794i)19-s + (−0.625 − 1.08i)23-s + (0.100 − 0.173i)25-s + (0.359 − 0.622i)31-s + (−0.821 − 1.42i)37-s + 0.312·41-s − 0.609·43-s + (−0.291 − 0.505i)47-s + (0.824 − 1.42i)53-s + 0.539·55-s + (−0.781 + 1.35i)59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 + 0.126i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.991 + 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3528\)    =    \(2^{3} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.991 + 0.126i$
Analytic conductor: \(28.1712\)
Root analytic conductor: \(5.30765\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3528} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3528,\ (\ :1/2),\ -0.991 + 0.126i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3209722599\)
\(L(\frac12)\) \(\approx\) \(0.3209722599\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (1 + 1.73i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (1 - 1.73i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 2T + 13T^{2} \)
17 \( 1 + (3 - 5.19i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2 - 3.46i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (3 + 5.19i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 + (-2 + 3.46i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (5 + 8.66i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 2T + 41T^{2} \)
43 \( 1 + 4T + 43T^{2} \)
47 \( 1 + (2 + 3.46i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-6 + 10.3i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (6 - 10.3i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3 + 5.19i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2 + 3.46i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 14T + 71T^{2} \)
73 \( 1 + (-1 + 1.73i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-4 - 6.92i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 16T + 83T^{2} \)
89 \( 1 + (-3 - 5.19i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 18T + 97T^{2} \)
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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

−8.356650412532347210833203915672, −7.64321550013826848539078481267, −6.68304848926556468219948100748, −5.96601860841122728004362357105, −5.13654557958103102800689401193, −4.21461812439350203914778916789, −3.80227330923544098251750680524, −2.42113287979179225495843421819, −1.47014701053706603693312788959, −0.097056364725263414226705901230, 1.38496393885218608056041179801, 2.85694728215589914817387002134, 3.18771686157930484868623176428, 4.31037158172445040616077874153, 5.14271294802645527691279459149, 5.98356369611582582772717245715, 6.89375100602734023606623402807, 7.29137050779089477964824882642, 8.165833121720289315957894878189, 8.890078517589631724907701632547

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