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Properties

Label 2-3528-392.237-c0-0-1
Degree $2$
Conductor $3528$
Sign $-0.518 + 0.855i$
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  + (0.781 − 0.623i)2-s + (0.222 − 0.974i)4-s + (1.40 − 0.678i)5-s + (−0.900 − 0.433i)7-s + (−0.433 − 0.900i)8-s + (0.678 − 1.40i)10-s + (−0.347 + 0.277i)11-s + (−0.974 + 0.222i)14-s + (−0.900 − 0.433i)16-s + (−0.347 − 1.52i)20-s + (−0.0990 + 0.433i)22-s + (0.900 − 1.12i)25-s + (−0.623 + 0.781i)28-s + (1.21 − 0.277i)29-s − 0.867i·31-s + (−0.974 + 0.222i)32-s + ⋯
L(s)  = 1  + (0.781 − 0.623i)2-s + (0.222 − 0.974i)4-s + (1.40 − 0.678i)5-s + (−0.900 − 0.433i)7-s + (−0.433 − 0.900i)8-s + (0.678 − 1.40i)10-s + (−0.347 + 0.277i)11-s + (−0.974 + 0.222i)14-s + (−0.900 − 0.433i)16-s + (−0.347 − 1.52i)20-s + (−0.0990 + 0.433i)22-s + (0.900 − 1.12i)25-s + (−0.623 + 0.781i)28-s + (1.21 − 0.277i)29-s − 0.867i·31-s + (−0.974 + 0.222i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.518 + 0.855i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.518 + 0.855i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3528\)    =    \(2^{3} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.518 + 0.855i$
Analytic conductor: \(1.76070\)
Root analytic conductor: \(1.32691\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3528} (2197, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3528,\ (\ :0),\ -0.518 + 0.855i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.171952912\)
\(L(\frac12)\) \(\approx\) \(2.171952912\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.781 + 0.623i)T \)
3 \( 1 \)
7 \( 1 + (0.900 + 0.433i)T \)
good5 \( 1 + (-1.40 + 0.678i)T + (0.623 - 0.781i)T^{2} \)
11 \( 1 + (0.347 - 0.277i)T + (0.222 - 0.974i)T^{2} \)
13 \( 1 + (-0.222 + 0.974i)T^{2} \)
17 \( 1 + (0.900 - 0.433i)T^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 + (-0.900 - 0.433i)T^{2} \)
29 \( 1 + (-1.21 + 0.277i)T + (0.900 - 0.433i)T^{2} \)
31 \( 1 + 0.867iT - T^{2} \)
37 \( 1 + (0.900 - 0.433i)T^{2} \)
41 \( 1 + (-0.623 + 0.781i)T^{2} \)
43 \( 1 + (-0.623 - 0.781i)T^{2} \)
47 \( 1 + (0.222 - 0.974i)T^{2} \)
53 \( 1 + (1.94 + 0.445i)T + (0.900 + 0.433i)T^{2} \)
59 \( 1 + (0.623 + 0.781i)T^{2} \)
61 \( 1 + (-0.900 + 0.433i)T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + (-0.900 - 0.433i)T^{2} \)
73 \( 1 + (-0.678 - 0.541i)T + (0.222 + 0.974i)T^{2} \)
79 \( 1 - 1.80T + T^{2} \)
83 \( 1 + (0.974 - 1.22i)T + (-0.222 - 0.974i)T^{2} \)
89 \( 1 + (0.222 + 0.974i)T^{2} \)
97 \( 1 - 1.94iT - T^{2} \)
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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

−8.805847328896577544887253832106, −7.68200124194065794901109760322, −6.50289385042061663547488448636, −6.27587746767191136135132286294, −5.34019103404792108693202875960, −4.76488956373981108087844693601, −3.83240502626080385870376767522, −2.81079470323309315630367626133, −2.07088450812452328883412299949, −0.969427434851349966664356510843, 1.94446946942997001313693331485, 2.89726385608736220457867865137, 3.28282801555582922826720016889, 4.64468430940047681004639783258, 5.42578958528331198284666173420, 6.11437165786796446910671364320, 6.50533902928826326471628487204, 7.18615161295430995015401001504, 8.198130178757124824523458604512, 8.978724314577500987225715561604

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