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Properties

Label 2-3960-440.109-c0-0-2
Degree $2$
Conductor $3960$
Sign $-i$
Analytic cond. $1.97629$
Root an. cond. $1.40580$
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 − i·5-s − 2·7-s − i·8-s + 10-s + i·11-s − 2i·14-s + 16-s + i·20-s − 22-s − 25-s + 2·28-s + 2·31-s + i·32-s + ⋯
L(s)  = 1  + i·2-s − 4-s − i·5-s − 2·7-s − i·8-s + 10-s + i·11-s − 2i·14-s + 16-s + i·20-s − 22-s − 25-s + 2·28-s + 2·31-s + i·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3960\)    =    \(2^{3} \cdot 3^{2} \cdot 5 \cdot 11\)
Sign: $-i$
Analytic conductor: \(1.97629\)
Root analytic conductor: \(1.40580\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3960} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3960,\ (\ :0),\ -i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7367839656\)
\(L(\frac12)\) \(\approx\) \(0.7367839656\)
\(L(1)\) 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 \( 1 \)
5 \( 1 + iT \)
11 \( 1 - iT \)
good7 \( 1 + 2T + T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 - 2T + T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 - 2iT - T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 - 2T + T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - 2iT - T^{2} \)
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

−8.812982482223015601622394317940, −8.069197698724880143985706539150, −7.23033937902870630758178903027, −6.58207218974627329522711917673, −6.02359901027962498270680217781, −5.18832742479788427609666569673, −4.37328754425237463248456331961, −3.71259277831989607516191508865, −2.60682931451682078377866780442, −0.899719447387035424950288718152, 0.56218169268051479198333510255, 2.27462517979186196670706891729, 3.19602483629408867223959706149, 3.30862581511211526808665064607, 4.32884756750620426229543927381, 5.58971411688820460309784520000, 6.28366043460594445186935420235, 6.76507104220192588980680516194, 7.86335525327547243594087859369, 8.635439420297348243659689296025

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