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Properties

Label 2-3920-980.379-c0-0-1
Degree $2$
Conductor $3920$
Sign $0.981 - 0.191i$
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  + (0.974 + 1.22i)3-s + (−0.623 − 0.781i)5-s + (−0.433 − 0.900i)7-s + (−0.321 + 1.40i)9-s + (0.347 − 1.52i)15-s + (0.678 − 1.40i)21-s + (1.40 − 0.678i)23-s + (−0.222 + 0.974i)25-s + (−0.626 + 0.301i)27-s + (1.62 + 0.781i)29-s + (−0.433 + 0.900i)35-s + (0.277 + 0.347i)41-s + (1.21 − 1.52i)43-s + (1.30 − 0.626i)45-s + (−0.347 − 1.52i)47-s + ⋯
L(s)  = 1  + (0.974 + 1.22i)3-s + (−0.623 − 0.781i)5-s + (−0.433 − 0.900i)7-s + (−0.321 + 1.40i)9-s + (0.347 − 1.52i)15-s + (0.678 − 1.40i)21-s + (1.40 − 0.678i)23-s + (−0.222 + 0.974i)25-s + (−0.626 + 0.301i)27-s + (1.62 + 0.781i)29-s + (−0.433 + 0.900i)35-s + (0.277 + 0.347i)41-s + (1.21 − 1.52i)43-s + (1.30 − 0.626i)45-s + (−0.347 − 1.52i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.981 - 0.191i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.981 - 0.191i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3920\)    =    \(2^{4} \cdot 5 \cdot 7^{2}\)
Sign: $0.981 - 0.191i$
Analytic conductor: \(1.95633\)
Root analytic conductor: \(1.39869\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3920} (1359, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3920,\ (\ :0),\ 0.981 - 0.191i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.581777741\)
\(L(\frac12)\) \(\approx\) \(1.581777741\)
\(L(1)\) 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 \( 1 + (0.623 + 0.781i)T \)
7 \( 1 + (0.433 + 0.900i)T \)
good3 \( 1 + (-0.974 - 1.22i)T + (-0.222 + 0.974i)T^{2} \)
11 \( 1 + (0.900 - 0.433i)T^{2} \)
13 \( 1 + (0.900 - 0.433i)T^{2} \)
17 \( 1 + (-0.623 - 0.781i)T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + (-1.40 + 0.678i)T + (0.623 - 0.781i)T^{2} \)
29 \( 1 + (-1.62 - 0.781i)T + (0.623 + 0.781i)T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + (-0.623 - 0.781i)T^{2} \)
41 \( 1 + (-0.277 - 0.347i)T + (-0.222 + 0.974i)T^{2} \)
43 \( 1 + (-1.21 + 1.52i)T + (-0.222 - 0.974i)T^{2} \)
47 \( 1 + (0.347 + 1.52i)T + (-0.900 + 0.433i)T^{2} \)
53 \( 1 + (-0.623 + 0.781i)T^{2} \)
59 \( 1 + (0.222 + 0.974i)T^{2} \)
61 \( 1 + (-1.12 - 0.541i)T + (0.623 + 0.781i)T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 + (-0.623 + 0.781i)T^{2} \)
73 \( 1 + (0.900 + 0.433i)T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + (0.193 - 0.846i)T + (-0.900 - 0.433i)T^{2} \)
89 \( 1 + (-0.400 + 1.75i)T + (-0.900 - 0.433i)T^{2} \)
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

−8.788567160338486700595767949247, −8.206167972496520249714818172561, −7.32815211800112659761177137838, −6.66323386808961985136699050092, −5.29363550954375689008540072473, −4.68678660592543122289401142687, −4.03300078292997248802920211074, −3.43285307602906641563689263457, −2.60861815263096424305820614122, −0.961935731378271305862598773930, 1.18709629076113864225720140445, 2.59131342209058089301815399992, 2.74744266875662243633022389374, 3.67543494628291151142067834195, 4.82735488813389744730206163470, 6.04161759349346416783980770226, 6.56668780983146546787400649353, 7.23446862382349705246633687305, 7.905985231756179633312035577387, 8.400928193779199091794282480438

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