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Properties

Label 2-3528-504.389-c0-0-0
Degree $2$
Conductor $3528$
Sign $0.458 + 0.888i$
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 + (0.258 + 0.965i)3-s − 4-s + (−0.965 + 1.67i)5-s + (−0.965 + 0.258i)6-s − i·8-s + (−0.866 + 0.499i)9-s + (−1.67 − 0.965i)10-s + (−0.258 − 0.965i)12-s + (−1.22 + 0.707i)13-s + (−1.86 − 0.500i)15-s + 16-s + (−0.499 − 0.866i)18-s + (−0.448 + 0.258i)19-s + (0.965 − 1.67i)20-s + ⋯
L(s)  = 1  + i·2-s + (0.258 + 0.965i)3-s − 4-s + (−0.965 + 1.67i)5-s + (−0.965 + 0.258i)6-s − i·8-s + (−0.866 + 0.499i)9-s + (−1.67 − 0.965i)10-s + (−0.258 − 0.965i)12-s + (−1.22 + 0.707i)13-s + (−1.86 − 0.500i)15-s + 16-s + (−0.499 − 0.866i)18-s + (−0.448 + 0.258i)19-s + (0.965 − 1.67i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.458 + 0.888i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.458 + 0.888i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3528\)    =    \(2^{3} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.458 + 0.888i$
Analytic conductor: \(1.76070\)
Root analytic conductor: \(1.32691\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3528} (2909, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3528,\ (\ :0),\ 0.458 + 0.888i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5461647850\)
\(L(\frac12)\) \(\approx\) \(0.5461647850\)
\(L(1)\) 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 \( 1 + (-0.258 - 0.965i)T \)
7 \( 1 \)
good5 \( 1 + (0.965 - 1.67i)T + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + (1.22 - 0.707i)T + (0.5 - 0.866i)T^{2} \)
17 \( 1 + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (0.448 - 0.258i)T + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (-0.5 - 0.866i)T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + (0.5 - 0.866i)T^{2} \)
41 \( 1 + (0.5 - 0.866i)T^{2} \)
43 \( 1 + (0.5 + 0.866i)T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + (-0.5 - 0.866i)T^{2} \)
59 \( 1 - 1.41T + T^{2} \)
61 \( 1 - 0.517iT - T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 - iT - T^{2} \)
73 \( 1 + (-0.5 - 0.866i)T^{2} \)
79 \( 1 + 1.73T + T^{2} \)
83 \( 1 + (-0.707 + 1.22i)T + (-0.5 - 0.866i)T^{2} \)
89 \( 1 + (0.5 - 0.866i)T^{2} \)
97 \( 1 + (-0.5 - 0.866i)T^{2} \)
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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

−9.274587215462706060594850032606, −8.530978993821457447837902194184, −7.68658036027040803989326001976, −7.19096819459350861096018028181, −6.61213567254692260931511658372, −5.62895437722675760730916096138, −4.73465674490419608823066934599, −4.03849582367480468640649145229, −3.33895036233477295349514455936, −2.52433929311177067181008101565, 0.34386840733286357329229250223, 1.19352398737308935427423036342, 2.34686414110017071426609793477, 3.23085070703288250942261952430, 4.24276308785334946261527189338, 4.97756789515569496346369295326, 5.49901000162888976158460977846, 6.88404471585556654565949079299, 7.75064717228099614434211115888, 8.218432549638604182074871172773

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