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Properties

Label 2-3528-3.2-c0-0-2
Degree $2$
Conductor $3528$
Sign $0.816 + 0.577i$
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  − 1.41i·5-s + 1.41i·11-s + 13-s + 19-s − 1.00·25-s + 31-s + 37-s − 1.41i·41-s − 43-s + 1.41i·47-s + 2.00·55-s − 1.41i·65-s − 67-s − 1.41i·71-s − 73-s + ⋯
L(s)  = 1  − 1.41i·5-s + 1.41i·11-s + 13-s + 19-s − 1.00·25-s + 31-s + 37-s − 1.41i·41-s − 43-s + 1.41i·47-s + 2.00·55-s − 1.41i·65-s − 67-s − 1.41i·71-s − 73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 + 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 + 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3528\)    =    \(2^{3} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.816 + 0.577i$
Analytic conductor: \(1.76070\)
Root analytic conductor: \(1.32691\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3528} (1961, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3528,\ (\ :0),\ 0.816 + 0.577i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.381835651\)
\(L(\frac12)\) \(\approx\) \(1.381835651\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + 1.41iT - T^{2} \)
11 \( 1 - 1.41iT - T^{2} \)
13 \( 1 - T + T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 - T + T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - T + T^{2} \)
37 \( 1 - T + T^{2} \)
41 \( 1 + 1.41iT - T^{2} \)
43 \( 1 + T + T^{2} \)
47 \( 1 - 1.41iT - T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 + T + T^{2} \)
71 \( 1 + 1.41iT - T^{2} \)
73 \( 1 + T + T^{2} \)
79 \( 1 - T + T^{2} \)
83 \( 1 + 1.41iT - T^{2} \)
89 \( 1 - T^{2} \)
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

−8.768616350758021268273154027414, −7.958255932286577027194084562947, −7.37068897313135167701915222835, −6.36507675851801021652450379983, −5.58059775696519935161552501208, −4.73605423315498264271410313792, −4.32489220316301974336206178468, −3.20932493005134807436263037457, −1.90357006909471318376401909340, −1.04792060629829524678440051001, 1.16721556874002780856124283411, 2.66713142168956078269495075562, 3.22848609756673137835300955363, 3.91796603982788771468442813246, 5.17956128824982557496189111535, 6.09628600542453795393486693217, 6.44055938447409992257991293426, 7.29924026205713056537978895911, 8.121974536911087717965107954350, 8.654997960599419851692348979964

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