Properties

Label 2-320-40.19-c2-0-14
Degree $2$
Conductor $320$
Sign $0.970 - 0.239i$
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  + 1.90i·3-s + (4.37 − 2.41i)5-s + 3.95·7-s + 5.35·9-s − 6.18·11-s + 4.94·13-s + (4.60 + 8.35i)15-s − 22.4i·17-s − 10.3·19-s + 7.54i·21-s + 39.6·23-s + (13.3 − 21.1i)25-s + 27.4i·27-s + 30.4i·29-s + 24.7i·31-s + ⋯
L(s)  = 1  + 0.636i·3-s + (0.875 − 0.482i)5-s + 0.565·7-s + 0.595·9-s − 0.562·11-s + 0.380·13-s + (0.306 + 0.557i)15-s − 1.31i·17-s − 0.546·19-s + 0.359i·21-s + 1.72·23-s + (0.534 − 0.845i)25-s + 1.01i·27-s + 1.04i·29-s + 0.797i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 - 0.239i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.970 - 0.239i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(320\)    =    \(2^{6} \cdot 5\)
Sign: $0.970 - 0.239i$
Analytic conductor: \(8.71936\)
Root analytic conductor: \(2.95285\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{320} (159, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 320,\ (\ :1),\ 0.970 - 0.239i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.10144 + 0.255257i\)
\(L(\frac12)\) \(\approx\) \(2.10144 + 0.255257i\)
\(L(2)\) 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 \( 1 + (-4.37 + 2.41i)T \)
good3 \( 1 - 1.90iT - 9T^{2} \)
7 \( 1 - 3.95T + 49T^{2} \)
11 \( 1 + 6.18T + 121T^{2} \)
13 \( 1 - 4.94T + 169T^{2} \)
17 \( 1 + 22.4iT - 289T^{2} \)
19 \( 1 + 10.3T + 361T^{2} \)
23 \( 1 - 39.6T + 529T^{2} \)
29 \( 1 - 30.4iT - 841T^{2} \)
31 \( 1 - 24.7iT - 961T^{2} \)
37 \( 1 - 24.0T + 1.36e3T^{2} \)
41 \( 1 - 31.0T + 1.68e3T^{2} \)
43 \( 1 - 52.2iT - 1.84e3T^{2} \)
47 \( 1 - 13.1T + 2.20e3T^{2} \)
53 \( 1 + 17.9T + 2.80e3T^{2} \)
59 \( 1 + 104.T + 3.48e3T^{2} \)
61 \( 1 + 57.2iT - 3.72e3T^{2} \)
67 \( 1 + 99.3iT - 4.48e3T^{2} \)
71 \( 1 + 16.7iT - 5.04e3T^{2} \)
73 \( 1 - 96.3iT - 5.32e3T^{2} \)
79 \( 1 + 139. iT - 6.24e3T^{2} \)
83 \( 1 + 91.7iT - 6.88e3T^{2} \)
89 \( 1 + 94.7T + 7.92e3T^{2} \)
97 \( 1 - 143. iT - 9.40e3T^{2} \)
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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

−11.13922753618171746128317760231, −10.56843726078667815627916761417, −9.474382721144677751939273917371, −8.952811773367064936340928818310, −7.67271700457406944637356274326, −6.52132190285628443682890587932, −5.10244170643936669412658190635, −4.70242280293951305533195325958, −2.96236720305406382801037191661, −1.34012864614716246248632412525, 1.39954541228058453633002029297, 2.53542953755352751332348217099, 4.24709195301158389483574866143, 5.63314115174638200839613527414, 6.51045155031312931023978916596, 7.48409132783519132263458860133, 8.426031976768956701932644027845, 9.595028547689538620055795980733, 10.57450206273392836311624333095, 11.18001890600374592879856006197

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