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Properties

Label 2-320-16.13-c1-0-3
Degree $2$
Conductor $320$
Sign $0.557 - 0.830i$
Analytic cond. $2.55521$
Root an. cond. $1.59850$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.32 + 2.32i)3-s + (0.707 − 0.707i)5-s − 0.982i·7-s + 7.82i·9-s + (1.62 − 1.62i)11-s + (−0.690 − 0.690i)13-s + 3.28·15-s − 2.19·17-s + (−1.92 − 1.92i)19-s + (2.28 − 2.28i)21-s + 2.01i·23-s − 1.00i·25-s + (−11.2 + 11.2i)27-s + (−5.27 − 5.27i)29-s − 0.435·31-s + ⋯
L(s)  = 1  + (1.34 + 1.34i)3-s + (0.316 − 0.316i)5-s − 0.371i·7-s + 2.60i·9-s + (0.490 − 0.490i)11-s + (−0.191 − 0.191i)13-s + 0.849·15-s − 0.532·17-s + (−0.441 − 0.441i)19-s + (0.498 − 0.498i)21-s + 0.420i·23-s − 0.200i·25-s + (−2.15 + 2.15i)27-s + (−0.978 − 0.978i)29-s − 0.0781·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.557 - 0.830i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.557 - 0.830i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(320\)    =    \(2^{6} \cdot 5\)
Sign: $0.557 - 0.830i$
Analytic conductor: \(2.55521\)
Root analytic conductor: \(1.59850\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{320} (81, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 320,\ (\ :1/2),\ 0.557 - 0.830i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.78166 + 0.950157i\)
\(L(\frac12)\) \(\approx\) \(1.78166 + 0.950157i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.707 + 0.707i)T \)
good3 \( 1 + (-2.32 - 2.32i)T + 3iT^{2} \)
7 \( 1 + 0.982iT - 7T^{2} \)
11 \( 1 + (-1.62 + 1.62i)T - 11iT^{2} \)
13 \( 1 + (0.690 + 0.690i)T + 13iT^{2} \)
17 \( 1 + 2.19T + 17T^{2} \)
19 \( 1 + (1.92 + 1.92i)T + 19iT^{2} \)
23 \( 1 - 2.01iT - 23T^{2} \)
29 \( 1 + (5.27 + 5.27i)T + 29iT^{2} \)
31 \( 1 + 0.435T + 31T^{2} \)
37 \( 1 + (5.79 - 5.79i)T - 37iT^{2} \)
41 \( 1 + 3.93iT - 41T^{2} \)
43 \( 1 + (-0.507 + 0.507i)T - 43iT^{2} \)
47 \( 1 - 9.21T + 47T^{2} \)
53 \( 1 + (-6.29 + 6.29i)T - 53iT^{2} \)
59 \( 1 + (-5.67 + 5.67i)T - 59iT^{2} \)
61 \( 1 + (3.60 + 3.60i)T + 61iT^{2} \)
67 \( 1 + (4.53 + 4.53i)T + 67iT^{2} \)
71 \( 1 - 10.3iT - 71T^{2} \)
73 \( 1 - 9.24iT - 73T^{2} \)
79 \( 1 + 15.4T + 79T^{2} \)
83 \( 1 + (-0.683 - 0.683i)T + 83iT^{2} \)
89 \( 1 + 5.44iT - 89T^{2} \)
97 \( 1 - 5.54T + 97T^{2} \)
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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

−11.50469487829456717190338453447, −10.55229597552234797182944133633, −9.812231921978649597133476114656, −8.974876629632275636956181151706, −8.401746163462147500016589119775, −7.21607424936166234516508225610, −5.55574143730971205646307758081, −4.40912153767375379151585791299, −3.58901982868440714644626810033, −2.26847662515630972629507063813, 1.70961435925483631530039717175, 2.64006209011260313467411738457, 3.95974038263984441374759071592, 5.91600465655867296964484728964, 6.93817077339998694128270177824, 7.52368578910402500039656226620, 8.814604844581719509380062168749, 9.127912757250589837672882681093, 10.45821353937183284835655903518, 11.88337152405043553905585572262

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