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Properties

Label 2-320-40.29-c5-0-31
Degree $2$
Conductor $320$
Sign $0.156 - 0.987i$
Analytic cond. $51.3228$
Root an. cond. $7.16399$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 28.5·3-s + (5.81 + 55.5i)5-s − 92.3i·7-s + 571.·9-s + 541. i·11-s − 782.·13-s + (165. + 1.58e3i)15-s + 1.55e3i·17-s + 484. i·19-s − 2.63e3i·21-s + 1.09e3i·23-s + (−3.05e3 + 646. i)25-s + 9.37e3·27-s − 597. i·29-s + 8.86e3·31-s + ⋯
L(s)  = 1  + 1.83·3-s + (0.104 + 0.994i)5-s − 0.712i·7-s + 2.35·9-s + 1.34i·11-s − 1.28·13-s + (0.190 + 1.82i)15-s + 1.30i·17-s + 0.307i·19-s − 1.30i·21-s + 0.432i·23-s + (−0.978 + 0.206i)25-s + 2.47·27-s − 0.131i·29-s + 1.65·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.156 - 0.987i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.156 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(320\)    =    \(2^{6} \cdot 5\)
Sign: $0.156 - 0.987i$
Analytic conductor: \(51.3228\)
Root analytic conductor: \(7.16399\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{320} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 320,\ (\ :5/2),\ 0.156 - 0.987i)\)

Particular Values

\(L(3)\) \(\approx\) \(3.868007521\)
\(L(\frac12)\) \(\approx\) \(3.868007521\)
\(L(\frac{7}{2})\) 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 \( 1 + (-5.81 - 55.5i)T \)
good3 \( 1 - 28.5T + 243T^{2} \)
7 \( 1 + 92.3iT - 1.68e4T^{2} \)
11 \( 1 - 541. iT - 1.61e5T^{2} \)
13 \( 1 + 782.T + 3.71e5T^{2} \)
17 \( 1 - 1.55e3iT - 1.41e6T^{2} \)
19 \( 1 - 484. iT - 2.47e6T^{2} \)
23 \( 1 - 1.09e3iT - 6.43e6T^{2} \)
29 \( 1 + 597. iT - 2.05e7T^{2} \)
31 \( 1 - 8.86e3T + 2.86e7T^{2} \)
37 \( 1 + 1.36e4T + 6.93e7T^{2} \)
41 \( 1 - 1.45e4T + 1.15e8T^{2} \)
43 \( 1 - 1.82e3T + 1.47e8T^{2} \)
47 \( 1 - 1.79e4iT - 2.29e8T^{2} \)
53 \( 1 - 1.65e3T + 4.18e8T^{2} \)
59 \( 1 + 1.44e4iT - 7.14e8T^{2} \)
61 \( 1 + 2.45e4iT - 8.44e8T^{2} \)
67 \( 1 + 2.68e4T + 1.35e9T^{2} \)
71 \( 1 - 4.80e4T + 1.80e9T^{2} \)
73 \( 1 + 6.06e4iT - 2.07e9T^{2} \)
79 \( 1 + 2.81e4T + 3.07e9T^{2} \)
83 \( 1 + 2.45e4T + 3.93e9T^{2} \)
89 \( 1 - 1.17e5T + 5.58e9T^{2} \)
97 \( 1 + 3.77e4iT - 8.58e9T^{2} \)
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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

−10.48087329934147501662113949149, −10.04020679563920659306973999053, −9.260640486139737492611621426806, −7.896891584604558178360631185552, −7.48052975350074382776969066759, −6.59031229369734477846376409367, −4.54592954243622555910061006149, −3.65894436412172762479760330864, −2.56491594168193819895649786278, −1.73901012137946667133582897178, 0.76026987547521405744777247544, 2.29755122281758943691043730992, 2.99996433560818239636333103323, 4.37337966428956333919092329454, 5.41216263832617348484514786157, 7.04131427113581820709176418073, 8.087493647720426780673735328425, 8.762870939417416834774477109786, 9.279621990884185580029660221123, 10.16758518459506080664078274793

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