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Properties

Label 2-3528-7.4-c1-0-40
Degree $2$
Conductor $3528$
Sign $0.266 + 0.963i$
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 + (2 − 3.46i)11-s + 2·13-s + (−1 + 1.73i)17-s + (−2 − 3.46i)19-s + (−4 − 6.92i)23-s + (0.500 − 0.866i)25-s − 6·29-s + (4 − 6.92i)31-s + (−3 − 5.19i)37-s − 6·41-s + 4·43-s + (−1 + 1.73i)53-s + 7.99·55-s + (−2 + 3.46i)59-s + ⋯
L(s)  = 1  + (0.447 + 0.774i)5-s + (0.603 − 1.04i)11-s + 0.554·13-s + (−0.242 + 0.420i)17-s + (−0.458 − 0.794i)19-s + (−0.834 − 1.44i)23-s + (0.100 − 0.173i)25-s − 1.11·29-s + (0.718 − 1.24i)31-s + (−0.493 − 0.854i)37-s − 0.937·41-s + 0.609·43-s + (−0.137 + 0.237i)53-s + 1.07·55-s + (−0.260 + 0.450i)59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.266 + 0.963i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.266 + 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3528\)    =    \(2^{3} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.266 + 0.963i$
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: \((2,\ 3528,\ (\ :1/2),\ 0.266 + 0.963i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.630172600\)
\(L(\frac12)\) \(\approx\) \(1.630172600\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2 + 3.46i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 2T + 13T^{2} \)
17 \( 1 + (1 - 1.73i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2 + 3.46i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (4 + 6.92i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 6T + 29T^{2} \)
31 \( 1 + (-4 + 6.92i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (3 + 5.19i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 6T + 41T^{2} \)
43 \( 1 - 4T + 43T^{2} \)
47 \( 1 + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (1 - 1.73i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (2 - 3.46i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1 + 1.73i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2 + 3.46i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 8T + 71T^{2} \)
73 \( 1 + (-5 + 8.66i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-4 - 6.92i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 4T + 83T^{2} \)
89 \( 1 + (-3 - 5.19i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 2T + 97T^{2} \)
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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

−8.523206547480919326328770445403, −7.71420777938584150005123227260, −6.65361833739716446526175153808, −6.29996019405270781317064729077, −5.65975145181356188248389810486, −4.42682838833444422738022997011, −3.73300437114305883411929180311, −2.75940405472030757182818077422, −1.94343588405441942659559806328, −0.47793955577332570900640088161, 1.36884072233065250071738913523, 1.89391777779346160987247761598, 3.33212284215746924064633671685, 4.13534507678828423403722059559, 4.97841640180916414662446311510, 5.64093376788795489009802645719, 6.49136920534852276201503134624, 7.22372983546029733968744825466, 8.058021560883418098093678656701, 8.806521443817236436082391013192

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