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Properties

Label 2-320-20.19-c2-0-0
Degree $2$
Conductor $320$
Sign $-0.950 + 0.309i$
Analytic cond. $8.71936$
Root an. cond. $2.95285$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5.30·3-s + (1.54 + 4.75i)5-s − 0.206·7-s + 19.1·9-s + 15.0i·11-s − 11.6i·13-s + (−8.20 − 25.2i)15-s + 18.1i·17-s − 19.3i·19-s + 1.09·21-s − 27.2·23-s + (−20.2 + 14.7i)25-s − 53.6·27-s − 44.4·29-s − 20.3i·31-s + ⋯
L(s)  = 1  − 1.76·3-s + (0.309 + 0.950i)5-s − 0.0295·7-s + 2.12·9-s + 1.36i·11-s − 0.899i·13-s + (−0.547 − 1.68i)15-s + 1.07i·17-s − 1.02i·19-s + 0.0521·21-s − 1.18·23-s + (−0.808 + 0.588i)25-s − 1.98·27-s − 1.53·29-s − 0.657i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.950 + 0.309i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.950 + 0.309i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(320\)    =    \(2^{6} \cdot 5\)
Sign: $-0.950 + 0.309i$
Analytic conductor: \(8.71936\)
Root analytic conductor: \(2.95285\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{320} (319, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 320,\ (\ :1),\ -0.950 + 0.309i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0242995 - 0.153140i\)
\(L(\frac12)\) \(\approx\) \(0.0242995 - 0.153140i\)
\(L(2)\) 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 \( 1 + (-1.54 - 4.75i)T \)
good3 \( 1 + 5.30T + 9T^{2} \)
7 \( 1 + 0.206T + 49T^{2} \)
11 \( 1 - 15.0iT - 121T^{2} \)
13 \( 1 + 11.6iT - 169T^{2} \)
17 \( 1 - 18.1iT - 289T^{2} \)
19 \( 1 + 19.3iT - 361T^{2} \)
23 \( 1 + 27.2T + 529T^{2} \)
29 \( 1 + 44.4T + 841T^{2} \)
31 \( 1 + 20.3iT - 961T^{2} \)
37 \( 1 + 18.1iT - 1.36e3T^{2} \)
41 \( 1 + 32.3T + 1.68e3T^{2} \)
43 \( 1 - 4.06T + 1.84e3T^{2} \)
47 \( 1 - 5.37T + 2.20e3T^{2} \)
53 \( 1 + 79.1iT - 2.80e3T^{2} \)
59 \( 1 - 83.3iT - 3.48e3T^{2} \)
61 \( 1 - 36.7T + 3.72e3T^{2} \)
67 \( 1 - 4.51T + 4.48e3T^{2} \)
71 \( 1 + 41.6iT - 5.04e3T^{2} \)
73 \( 1 + 41.5iT - 5.32e3T^{2} \)
79 \( 1 - 15.5iT - 6.24e3T^{2} \)
83 \( 1 - 50.9T + 6.88e3T^{2} \)
89 \( 1 - 10.8T + 7.92e3T^{2} \)
97 \( 1 - 12.1iT - 9.40e3T^{2} \)
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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

−11.76753584549407430454705753550, −10.97170299636923883384081715978, −10.28845904033521613896597338952, −9.647348085310738614601218500662, −7.72229635869875988218420744337, −6.87358735133603236790410654649, −6.05053918826621306885287254838, −5.20477975635770378624986351493, −3.95302099623070550227117488543, −1.96051157650938159502288612661, 0.094126798866416645125565975325, 1.47140267645471853720847170583, 3.97550227200125065243422161662, 5.13628966251577395994207708136, 5.79913145896533700313206151316, 6.63222995194275488911783256559, 7.992222432407907736467802786876, 9.227243340145671647944952912720, 10.08935626443324681009782186206, 11.16712651133376455976058404124

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