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Properties

Label 2-320-80.53-c2-0-8
Degree $2$
Conductor $320$
Sign $0.902 + 0.430i$
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  − 2.77·3-s + (−2.05 + 4.55i)5-s + (−5.39 − 5.39i)7-s − 1.30·9-s + (−2.98 − 2.98i)11-s + 21.1·13-s + (5.70 − 12.6i)15-s + (6.66 − 6.66i)17-s + (14.3 + 14.3i)19-s + (14.9 + 14.9i)21-s + (2.07 − 2.07i)23-s + (−16.5 − 18.7i)25-s + 28.5·27-s + (−17.5 − 17.5i)29-s + 23.6·31-s + ⋯
L(s)  = 1  − 0.924·3-s + (−0.411 + 0.911i)5-s + (−0.770 − 0.770i)7-s − 0.145·9-s + (−0.270 − 0.270i)11-s + 1.63·13-s + (0.380 − 0.842i)15-s + (0.392 − 0.392i)17-s + (0.753 + 0.753i)19-s + (0.712 + 0.712i)21-s + (0.0901 − 0.0901i)23-s + (−0.661 − 0.750i)25-s + 1.05·27-s + (−0.604 − 0.604i)29-s + 0.762·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.902 + 0.430i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.902 + 0.430i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(320\)    =    \(2^{6} \cdot 5\)
Sign: $0.902 + 0.430i$
Analytic conductor: \(8.71936\)
Root analytic conductor: \(2.95285\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{320} (113, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 320,\ (\ :1),\ 0.902 + 0.430i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.873150 - 0.197551i\)
\(L(\frac12)\) \(\approx\) \(0.873150 - 0.197551i\)
\(L(2)\) 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 \( 1 + (2.05 - 4.55i)T \)
good3 \( 1 + 2.77T + 9T^{2} \)
7 \( 1 + (5.39 + 5.39i)T + 49iT^{2} \)
11 \( 1 + (2.98 + 2.98i)T + 121iT^{2} \)
13 \( 1 - 21.1T + 169T^{2} \)
17 \( 1 + (-6.66 + 6.66i)T - 289iT^{2} \)
19 \( 1 + (-14.3 - 14.3i)T + 361iT^{2} \)
23 \( 1 + (-2.07 + 2.07i)T - 529iT^{2} \)
29 \( 1 + (17.5 + 17.5i)T + 841iT^{2} \)
31 \( 1 - 23.6T + 961T^{2} \)
37 \( 1 - 65.3T + 1.36e3T^{2} \)
41 \( 1 - 1.18iT - 1.68e3T^{2} \)
43 \( 1 + 9.57iT - 1.84e3T^{2} \)
47 \( 1 + (-47.2 + 47.2i)T - 2.20e3iT^{2} \)
53 \( 1 + 99.2iT - 2.80e3T^{2} \)
59 \( 1 + (54.7 - 54.7i)T - 3.48e3iT^{2} \)
61 \( 1 + (12.1 - 12.1i)T - 3.72e3iT^{2} \)
67 \( 1 - 109. iT - 4.48e3T^{2} \)
71 \( 1 + 73.1iT - 5.04e3T^{2} \)
73 \( 1 + (17.0 - 17.0i)T - 5.32e3iT^{2} \)
79 \( 1 - 2.15iT - 6.24e3T^{2} \)
83 \( 1 + 76.4T + 6.88e3T^{2} \)
89 \( 1 - 38.7T + 7.92e3T^{2} \)
97 \( 1 + (-8.53 + 8.53i)T - 9.40e3iT^{2} \)
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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

−11.31954531681817160237325737033, −10.58013288996158072753183230366, −9.841539403462627153507760818578, −8.367006730427826920073810150855, −7.32875623500152433083356078512, −6.33297665633291101753162990153, −5.71375787440411221597328869696, −4.01125151297376368082621164243, −3.08458271401865603597025540228, −0.66038839878458387323898195773, 0.978534371823840289810737950271, 3.12225214684480020699052645896, 4.55667649575010969913117422113, 5.70078626519248657792927897033, 6.21393843956466490858789759690, 7.72177814511362049021510025603, 8.791156664266779580420085613637, 9.461158700280176538516189916318, 10.84349057402293358510318832532, 11.49102399290366253349742923488

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