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Properties

Label 2-2320-580.219-c0-0-3
Degree $2$
Conductor $2320$
Sign $0.855 - 0.517i$
Analytic cond. $1.15783$
Root an. cond. $1.07602$
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  + (−0.541 + 0.678i)3-s + (0.900 + 0.433i)5-s + (0.974 − 1.22i)7-s + (0.0549 + 0.240i)9-s + (−0.781 + 0.376i)15-s + (0.301 + 1.32i)21-s + (0.781 − 0.376i)23-s + (0.623 + 0.781i)25-s + (−0.974 − 0.469i)27-s + (0.623 − 0.781i)29-s + (1.40 − 0.678i)35-s + 0.445·41-s + (−1.75 + 0.846i)43-s + (−0.0549 + 0.240i)45-s + (0.347 − 1.52i)47-s + ⋯
L(s)  = 1  + (−0.541 + 0.678i)3-s + (0.900 + 0.433i)5-s + (0.974 − 1.22i)7-s + (0.0549 + 0.240i)9-s + (−0.781 + 0.376i)15-s + (0.301 + 1.32i)21-s + (0.781 − 0.376i)23-s + (0.623 + 0.781i)25-s + (−0.974 − 0.469i)27-s + (0.623 − 0.781i)29-s + (1.40 − 0.678i)35-s + 0.445·41-s + (−1.75 + 0.846i)43-s + (−0.0549 + 0.240i)45-s + (0.347 − 1.52i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.855 - 0.517i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.855 - 0.517i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2320\)    =    \(2^{4} \cdot 5 \cdot 29\)
Sign: $0.855 - 0.517i$
Analytic conductor: \(1.15783\)
Root analytic conductor: \(1.07602\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2320} (799, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2320,\ (\ :0),\ 0.855 - 0.517i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.336317328\)
\(L(\frac12)\) \(\approx\) \(1.336317328\)
\(L(1)\) 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 \( 1 + (-0.900 - 0.433i)T \)
29 \( 1 + (-0.623 + 0.781i)T \)
good3 \( 1 + (0.541 - 0.678i)T + (-0.222 - 0.974i)T^{2} \)
7 \( 1 + (-0.974 + 1.22i)T + (-0.222 - 0.974i)T^{2} \)
11 \( 1 + (0.900 + 0.433i)T^{2} \)
13 \( 1 + (0.900 + 0.433i)T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 + (0.222 - 0.974i)T^{2} \)
23 \( 1 + (-0.781 + 0.376i)T + (0.623 - 0.781i)T^{2} \)
31 \( 1 + (-0.623 - 0.781i)T^{2} \)
37 \( 1 + (0.900 - 0.433i)T^{2} \)
41 \( 1 - 0.445T + T^{2} \)
43 \( 1 + (1.75 - 0.846i)T + (0.623 - 0.781i)T^{2} \)
47 \( 1 + (-0.347 + 1.52i)T + (-0.900 - 0.433i)T^{2} \)
53 \( 1 + (-0.623 - 0.781i)T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + (1.24 - 1.56i)T + (-0.222 - 0.974i)T^{2} \)
67 \( 1 + (-0.900 + 0.433i)T^{2} \)
71 \( 1 + (0.900 + 0.433i)T^{2} \)
73 \( 1 + (-0.623 + 0.781i)T^{2} \)
79 \( 1 + (0.900 - 0.433i)T^{2} \)
83 \( 1 + (-1.21 - 1.52i)T + (-0.222 + 0.974i)T^{2} \)
89 \( 1 + (1.12 + 0.541i)T + (0.623 + 0.781i)T^{2} \)
97 \( 1 + (0.222 - 0.974i)T^{2} \)
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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

−9.481510769526061316349352004499, −8.476453476161558507555513525457, −7.64051740197154363766467773644, −6.91582424828178982498228438662, −6.07380339542690314132480463801, −5.10405836346289521559436615915, −4.65767799864013804810367681297, −3.72332201843994407407997748446, −2.44313815764331440093304674331, −1.29822828356425579814252719427, 1.28559529555366955272867843520, 2.00453247062659773354342850098, 3.13262712524698575902515676491, 4.70433871183854443832243995617, 5.27016864370105111746237817977, 5.97782519051321198790815570001, 6.60630397641935250875955856762, 7.55640756052351687242137440293, 8.480042411354167416424377698141, 9.038594642662521338949495813341

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