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Properties

Label 2-3920-140.79-c0-0-8
Degree $2$
Conductor $3920$
Sign $0.386 + 0.922i$
Analytic cond. $1.95633$
Root an. cond. $1.39869$
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  + (0.5 − 0.866i)5-s + (0.5 − 0.866i)9-s + (−0.499 − 0.866i)25-s + 2·29-s − 2·41-s + (−0.499 − 0.866i)45-s + (1 − 1.73i)61-s + (−0.499 − 0.866i)81-s + (−1 + 1.73i)89-s + (−1 − 1.73i)101-s + (1 + 1.73i)109-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)5-s + (0.5 − 0.866i)9-s + (−0.499 − 0.866i)25-s + 2·29-s − 2·41-s + (−0.499 − 0.866i)45-s + (1 − 1.73i)61-s + (−0.499 − 0.866i)81-s + (−1 + 1.73i)89-s + (−1 − 1.73i)101-s + (1 + 1.73i)109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3920\)    =    \(2^{4} \cdot 5 \cdot 7^{2}\)
Sign: $0.386 + 0.922i$
Analytic conductor: \(1.95633\)
Root analytic conductor: \(1.39869\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3920} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3920,\ (\ :0),\ 0.386 + 0.922i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.458337947\)
\(L(\frac12)\) \(\approx\) \(1.458337947\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 \)
good3 \( 1 + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (-0.5 - 0.866i)T^{2} \)
29 \( 1 - 2T + T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 + (0.5 + 0.866i)T^{2} \)
41 \( 1 + 2T + T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + (-0.5 - 0.866i)T^{2} \)
53 \( 1 + (0.5 - 0.866i)T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.5 + 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (0.5 - 0.866i)T^{2} \)
79 \( 1 + (0.5 + 0.866i)T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \)
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

−8.544904331268213053730215793229, −7.976548706184446865390362053358, −6.78459089254537512914890994565, −6.45892390804888911059095724372, −5.43929766255881672517468222856, −4.78245619302691252748903274452, −3.99719055258620313241446925148, −3.03745872317234705303229426957, −1.85906446390295164884588257946, −0.883798214941200687861064139717, 1.48812474695699962345333133290, 2.43850170423643638790939319817, 3.19217669200412904210876527403, 4.27086280084149204760307857238, 5.06647583653431159683539426337, 5.84483935684594269645001444564, 6.75185943142755218652918484211, 7.12687349369153388765452661726, 8.076981934288995250842476762173, 8.658324089163570759377829810949

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