Properties

Label 2-3920-35.19-c0-0-0
Degree $2$
Conductor $3920$
Sign $0.895 + 0.444i$
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)3-s + (0.5 − 0.866i)5-s + (−0.5 − 0.866i)11-s + 13-s + 0.999·15-s + (−0.5 − 0.866i)17-s + (−0.499 − 0.866i)25-s + 27-s − 29-s + (0.499 − 0.866i)33-s + (0.5 + 0.866i)39-s + (0.5 − 0.866i)47-s + (0.499 − 0.866i)51-s − 0.999·55-s + (0.5 − 0.866i)65-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)3-s + (0.5 − 0.866i)5-s + (−0.5 − 0.866i)11-s + 13-s + 0.999·15-s + (−0.5 − 0.866i)17-s + (−0.499 − 0.866i)25-s + 27-s − 29-s + (0.499 − 0.866i)33-s + (0.5 + 0.866i)39-s + (0.5 − 0.866i)47-s + (0.499 − 0.866i)51-s − 0.999·55-s + (0.5 − 0.866i)65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.895 + 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.895 + 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3920\)    =    \(2^{4} \cdot 5 \cdot 7^{2}\)
Sign: $0.895 + 0.444i$
Analytic conductor: \(1.95633\)
Root analytic conductor: \(1.39869\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3920} (1489, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3920,\ (\ :0),\ 0.895 + 0.444i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.681862900\)
\(L(\frac12)\) \(\approx\) \(1.681862900\)
\(L(1)\) 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 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
13 \( 1 - T + T^{2} \)
17 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 + T + T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 + (0.5 + 0.866i)T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
53 \( 1 + (0.5 - 0.866i)T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T^{2} \)
71 \( 1 + 2T + T^{2} \)
73 \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 - 2T + T^{2} \)
89 \( 1 + (0.5 + 0.866i)T^{2} \)
97 \( 1 - T + T^{2} \)
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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

−8.733614783660548093633074642694, −8.212454549756463983408855911705, −7.17316743724677732856886375801, −6.18882148716012650976353155581, −5.52467628232886686018977425399, −4.78707212293264494304231723380, −3.97712806400171567523942674410, −3.27408788186626391917586658343, −2.24601110414887581424762709628, −0.927869125450152359380604797974, 1.59228024958307104524064297409, 2.13371016779172160361029019879, 3.04221483719978103522570925985, 3.97032340608221986595279972084, 5.00004235517666970918218422334, 6.07442689358017728000071771267, 6.46720430206765895279857520882, 7.46690627729885636446035196478, 7.65222500137946550544495172951, 8.652446336320477945843565088367

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