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Properties

Label 2-3528-56.45-c0-0-4
Degree $2$
Conductor $3528$
Sign $-0.553 + 0.832i$
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 + (0.499 − 0.866i)4-s − 0.999i·8-s + (−1.73 − i)11-s + (−0.5 − 0.866i)16-s − 1.99·22-s + (0.5 − 0.866i)25-s − 2i·29-s + (−0.866 − 0.499i)32-s + (−1.73 + 0.999i)44-s − 0.999i·50-s + (1.73 + i)53-s + (−1 − 1.73i)58-s − 0.999·64-s + (1 + 1.73i)79-s + ⋯
L(s)  = 1  + (0.866 − 0.5i)2-s + (0.499 − 0.866i)4-s − 0.999i·8-s + (−1.73 − i)11-s + (−0.5 − 0.866i)16-s − 1.99·22-s + (0.5 − 0.866i)25-s − 2i·29-s + (−0.866 − 0.499i)32-s + (−1.73 + 0.999i)44-s − 0.999i·50-s + (1.73 + i)53-s + (−1 − 1.73i)58-s − 0.999·64-s + (1 + 1.73i)79-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.553 + 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.553 + 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3528\)    =    \(2^{3} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.553 + 0.832i$
Analytic conductor: \(1.76070\)
Root analytic conductor: \(1.32691\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3528} (325, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3528,\ (\ :0),\ -0.553 + 0.832i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.733295606\)
\(L(\frac12)\) \(\approx\) \(1.733295606\)
\(L(1)\) 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 \( 1 \)
7 \( 1 \)
good5 \( 1 + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (1.73 + i)T + (0.5 + 0.866i)T^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + 2iT - T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 + (0.5 - 0.866i)T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (0.5 - 0.866i)T^{2} \)
53 \( 1 + (-1.73 - i)T + (0.5 + 0.866i)T^{2} \)
59 \( 1 + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (0.5 + 0.866i)T^{2} \)
79 \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \)
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

−8.362165157640310453146580533686, −7.80409036854896822833728878440, −6.84138361986475909030658450694, −5.95028610396301907024631845044, −5.49350484176038121036309887863, −4.64090549380733303652997222573, −3.82766418044223564874127557290, −2.79268207423488883998197423758, −2.32975314150966046371305273841, −0.71351141845824042700881767578, 1.88294125771003227164373178683, 2.79658206931692803497334716176, 3.57013175903297849591344441358, 4.71314553698536924946225385993, 5.13917033434594768298071814951, 5.81247766232003993913943929622, 7.02765788639734246727366575542, 7.20990963633034751971515585818, 8.097936472207345714948315721082, 8.751123487657733639730374166064

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