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Properties

Label 2-3528-56.13-c0-0-1
Degree $2$
Conductor $3528$
Sign $-0.654 + 0.755i$
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  − i·2-s − 4-s − 1.73·5-s + i·8-s + 1.73i·10-s + i·11-s + 16-s + 1.73·20-s + 22-s + 1.99·25-s − i·29-s − 1.73i·31-s − i·32-s − 1.73i·40-s − i·44-s + ⋯
L(s)  = 1  − i·2-s − 4-s − 1.73·5-s + i·8-s + 1.73i·10-s + i·11-s + 16-s + 1.73·20-s + 22-s + 1.99·25-s − i·29-s − 1.73i·31-s − i·32-s − 1.73i·40-s − i·44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.654 + 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.654 + 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3528\)    =    \(2^{3} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.654 + 0.755i$
Analytic conductor: \(1.76070\)
Root analytic conductor: \(1.32691\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3528} (685, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3528,\ (\ :0),\ -0.654 + 0.755i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5717469621\)
\(L(\frac12)\) \(\approx\) \(0.5717469621\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + 1.73T + T^{2} \)
11 \( 1 - iT - T^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 + iT - T^{2} \)
31 \( 1 + 1.73iT - T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + iT - T^{2} \)
59 \( 1 + 1.73T + T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 - T + T^{2} \)
83 \( 1 - 1.73T + T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + 1.73iT - T^{2} \)
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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

−8.438698510785410031078001938741, −7.82647678313390958979881922868, −7.36478488934704653931739490302, −6.22697295273589252441928874691, −5.03903684800058156416594910986, −4.34073256674451001629812577321, −3.87728776829330413822772124634, −2.96408638745224397027568615899, −1.93914934136199384104800308841, −0.44221936371038336493023303586, 0.993200744814707287351114541534, 3.18061693015274264072762620974, 3.61052633163461439627682900120, 4.56781568875849452329169665779, 5.16942615048672197712307958247, 6.19977681486935614640579465977, 6.92408754919693638213430329843, 7.58122414654573459107572826813, 8.173216074995425293347818336289, 8.734622149813180270690000644312

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