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Properties

Label 2-3528-504.331-c0-0-5
Degree $2$
Conductor $3528$
Sign $-0.823 + 0.566i$
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 − 3-s + (−0.499 + 0.866i)4-s + (0.5 + 0.866i)6-s + 0.999·8-s + 9-s − 11-s + (0.499 − 0.866i)12-s + (−0.5 − 0.866i)16-s + (−0.5 − 0.866i)17-s + (−0.5 − 0.866i)18-s + (−0.5 + 0.866i)19-s + (0.5 + 0.866i)22-s − 0.999·24-s + 25-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)2-s − 3-s + (−0.499 + 0.866i)4-s + (0.5 + 0.866i)6-s + 0.999·8-s + 9-s − 11-s + (0.499 − 0.866i)12-s + (−0.5 − 0.866i)16-s + (−0.5 − 0.866i)17-s + (−0.5 − 0.866i)18-s + (−0.5 + 0.866i)19-s + (0.5 + 0.866i)22-s − 0.999·24-s + 25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.823 + 0.566i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.823 + 0.566i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3528\)    =    \(2^{3} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.823 + 0.566i$
Analytic conductor: \(1.76070\)
Root analytic conductor: \(1.32691\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3528} (1843, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3528,\ (\ :0),\ -0.823 + 0.566i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3745718142\)
\(L(\frac12)\) \(\approx\) \(0.3745718142\)
\(L(1)\) 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 \( 1 + T \)
7 \( 1 \)
good5 \( 1 - T^{2} \)
11 \( 1 + T + T^{2} \)
13 \( 1 + (0.5 - 0.866i)T^{2} \)
17 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + (0.5 + 0.866i)T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 + (0.5 + 0.866i)T^{2} \)
41 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
43 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
47 \( 1 + (0.5 - 0.866i)T^{2} \)
53 \( 1 + (0.5 - 0.866i)T^{2} \)
59 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
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 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \)
89 \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \)
97 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
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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

−8.616606714663420952236495605054, −7.66860064383304134040964046895, −7.16363099557469327309112437386, −6.20142796470599633613551456457, −5.21368660857939697154599063451, −4.66696738729661584203730275865, −3.73360089985589739853998253084, −2.67578245681792176034752873628, −1.70046031026273636474131648898, −0.34225332226001629267785543443, 1.12500370143614322942335816490, 2.42071744323730659207743186455, 3.99711758633983183236691015860, 4.87396443684704615000062290957, 5.28213576482073163866129744887, 6.33164101488568298723070583264, 6.61697613230895577108834359555, 7.53475555715532985510426433076, 8.179716970705391039201441908813, 8.960015034139428585056379756583

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