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Properties

Label 2-3528-504.419-c0-0-4
Degree $2$
Conductor $3528$
Sign $0.328 + 0.944i$
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 + (−0.608 + 0.793i)3-s + (0.499 − 0.866i)4-s + (−0.130 + 0.991i)6-s − 0.999i·8-s + (−0.258 − 0.965i)9-s + (0.448 − 0.258i)11-s + (0.382 + 0.923i)12-s + (−0.5 − 0.866i)16-s − 0.261·17-s + (−0.707 − 0.707i)18-s − 1.21i·19-s + (0.258 − 0.448i)22-s + (0.793 + 0.608i)24-s + (−0.5 − 0.866i)25-s + ⋯
L(s)  = 1  + (0.866 − 0.5i)2-s + (−0.608 + 0.793i)3-s + (0.499 − 0.866i)4-s + (−0.130 + 0.991i)6-s − 0.999i·8-s + (−0.258 − 0.965i)9-s + (0.448 − 0.258i)11-s + (0.382 + 0.923i)12-s + (−0.5 − 0.866i)16-s − 0.261·17-s + (−0.707 − 0.707i)18-s − 1.21i·19-s + (0.258 − 0.448i)22-s + (0.793 + 0.608i)24-s + (−0.5 − 0.866i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.328 + 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.328 + 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3528\)    =    \(2^{3} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.328 + 0.944i$
Analytic conductor: \(1.76070\)
Root analytic conductor: \(1.32691\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3528} (2939, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3528,\ (\ :0),\ 0.328 + 0.944i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.651106465\)
\(L(\frac12)\) \(\approx\) \(1.651106465\)
\(L(1)\) 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 \( 1 + (0.608 - 0.793i)T \)
7 \( 1 \)
good5 \( 1 + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (-0.448 + 0.258i)T + (0.5 - 0.866i)T^{2} \)
13 \( 1 + (-0.5 - 0.866i)T^{2} \)
17 \( 1 + 0.261T + T^{2} \)
19 \( 1 + 1.21iT - T^{2} \)
23 \( 1 + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + (-0.5 + 0.866i)T^{2} \)
31 \( 1 + (-0.5 - 0.866i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (-0.991 + 1.71i)T + (-0.5 - 0.866i)T^{2} \)
43 \( 1 + (-0.965 - 1.67i)T + (-0.5 + 0.866i)T^{2} \)
47 \( 1 + (0.5 - 0.866i)T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + (0.793 - 1.37i)T + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + 1.58iT - T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (-0.923 - 1.60i)T + (-0.5 + 0.866i)T^{2} \)
89 \( 1 + 0.765T + T^{2} \)
97 \( 1 + (-1.71 + 0.991i)T + (0.5 - 0.866i)T^{2} \)
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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

−9.100060770893776527377636549956, −7.74050231974693315020713797057, −6.72020169953726013940087383748, −6.18831303626625527541751931173, −5.46977134059834355132715305143, −4.61726608430342488731325078144, −4.13746799787339816671148761662, −3.23124673638630604726993368348, −2.31076065634306174901957810517, −0.806895941818325548827357135264, 1.52148995734800611294121360585, 2.44935418868777478531954607992, 3.60413677162114432921493426430, 4.40935877704132894184945400678, 5.32769131800438212687247631222, 5.91844457446402600477262290491, 6.55729264847731575263983337785, 7.29468455257523598720359969965, 7.84324123980901790555764012315, 8.561441179639000933681374879404

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