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Properties

Label 2-3528-392.45-c0-0-1
Degree $2$
Conductor $3528$
Sign $-0.138 + 0.990i$
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  + (0.563 − 0.826i)2-s + (−0.365 − 0.930i)4-s + (1.90 − 0.587i)5-s + (0.955 + 0.294i)7-s + (−0.974 − 0.222i)8-s + (0.587 − 1.90i)10-s + (−0.728 − 0.0546i)11-s + (0.781 − 0.623i)14-s + (−0.733 + 0.680i)16-s + (−1.24 − 1.55i)20-s + (−0.455 + 0.571i)22-s + (2.46 − 1.67i)25-s + (−0.0747 − 0.997i)28-s + (−1.29 + 1.03i)29-s + (−1.17 − 0.680i)31-s + (0.149 + 0.988i)32-s + ⋯
L(s)  = 1  + (0.563 − 0.826i)2-s + (−0.365 − 0.930i)4-s + (1.90 − 0.587i)5-s + (0.955 + 0.294i)7-s + (−0.974 − 0.222i)8-s + (0.587 − 1.90i)10-s + (−0.728 − 0.0546i)11-s + (0.781 − 0.623i)14-s + (−0.733 + 0.680i)16-s + (−1.24 − 1.55i)20-s + (−0.455 + 0.571i)22-s + (2.46 − 1.67i)25-s + (−0.0747 − 0.997i)28-s + (−1.29 + 1.03i)29-s + (−1.17 − 0.680i)31-s + (0.149 + 0.988i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.138 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.138 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3528\)    =    \(2^{3} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.138 + 0.990i$
Analytic conductor: \(1.76070\)
Root analytic conductor: \(1.32691\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3528} (829, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3528,\ (\ :0),\ -0.138 + 0.990i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.363376243\)
\(L(\frac12)\) \(\approx\) \(2.363376243\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.563 + 0.826i)T \)
3 \( 1 \)
7 \( 1 + (-0.955 - 0.294i)T \)
good5 \( 1 + (-1.90 + 0.587i)T + (0.826 - 0.563i)T^{2} \)
11 \( 1 + (0.728 + 0.0546i)T + (0.988 + 0.149i)T^{2} \)
13 \( 1 + (0.623 - 0.781i)T^{2} \)
17 \( 1 + (-0.955 + 0.294i)T^{2} \)
19 \( 1 + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.955 + 0.294i)T^{2} \)
29 \( 1 + (1.29 - 1.03i)T + (0.222 - 0.974i)T^{2} \)
31 \( 1 + (1.17 + 0.680i)T + (0.5 + 0.866i)T^{2} \)
37 \( 1 + (0.733 + 0.680i)T^{2} \)
41 \( 1 + (0.900 + 0.433i)T^{2} \)
43 \( 1 + (0.900 - 0.433i)T^{2} \)
47 \( 1 + (-0.365 - 0.930i)T^{2} \)
53 \( 1 + (-0.930 + 0.365i)T + (0.733 - 0.680i)T^{2} \)
59 \( 1 + (1.65 + 0.510i)T + (0.826 + 0.563i)T^{2} \)
61 \( 1 + (-0.733 - 0.680i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T^{2} \)
71 \( 1 + (-0.222 - 0.974i)T^{2} \)
73 \( 1 + (-1.09 - 1.61i)T + (-0.365 + 0.930i)T^{2} \)
79 \( 1 + (-0.955 - 1.65i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + (-1.01 - 0.488i)T + (0.623 + 0.781i)T^{2} \)
89 \( 1 + (0.988 - 0.149i)T^{2} \)
97 \( 1 + 0.298iT - T^{2} \)
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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

−8.900756883814366776422003316778, −8.014950695211242142422041339946, −6.81081122811371686108426144719, −5.88043685769564181843660490816, −5.28404879890855584254272869850, −5.07684355169822916418217755699, −3.91721657368873807400019036352, −2.62355283374400899319938575907, −2.02920177195495647505728727316, −1.29315290256264806755649089514, 1.76856937188771243555789199169, 2.51243047076713129416461380344, 3.53346963718979461689515144685, 4.70663209760126008080265623290, 5.36451666921070893948002571436, 5.84553120752546074757206157983, 6.60669496503500031304911951051, 7.40156898159973342179762290003, 7.935867902727873589698101948242, 9.064245281860659499326696078555

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