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Properties

Label 2-320-80.3-c1-0-9
Degree $2$
Conductor $320$
Sign $-0.361 + 0.932i$
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  − 0.692·3-s + (−0.245 − 2.22i)5-s + (0.343 − 0.343i)7-s − 2.52·9-s + (−0.843 − 0.843i)11-s − 3.68i·13-s + (0.169 + 1.53i)15-s + (0.412 − 0.412i)17-s + (−5.37 − 5.37i)19-s + (−0.238 + 0.238i)21-s + (3.08 + 3.08i)23-s + (−4.87 + 1.09i)25-s + 3.82·27-s + (4.22 − 4.22i)29-s − 8.75i·31-s + ⋯
L(s)  = 1  − 0.399·3-s + (−0.109 − 0.993i)5-s + (0.129 − 0.129i)7-s − 0.840·9-s + (−0.254 − 0.254i)11-s − 1.02i·13-s + (0.0438 + 0.397i)15-s + (0.0999 − 0.0999i)17-s + (−1.23 − 1.23i)19-s + (−0.0519 + 0.0519i)21-s + (0.643 + 0.643i)23-s + (−0.975 + 0.218i)25-s + 0.735·27-s + (0.785 − 0.785i)29-s − 1.57i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.361 + 0.932i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.361 + 0.932i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(320\)    =    \(2^{6} \cdot 5\)
Sign: $-0.361 + 0.932i$
Analytic conductor: \(2.55521\)
Root analytic conductor: \(1.59850\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{320} (303, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 320,\ (\ :1/2),\ -0.361 + 0.932i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.465864 - 0.680586i\)
\(L(\frac12)\) \(\approx\) \(0.465864 - 0.680586i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.245 + 2.22i)T \)
good3 \( 1 + 0.692T + 3T^{2} \)
7 \( 1 + (-0.343 + 0.343i)T - 7iT^{2} \)
11 \( 1 + (0.843 + 0.843i)T + 11iT^{2} \)
13 \( 1 + 3.68iT - 13T^{2} \)
17 \( 1 + (-0.412 + 0.412i)T - 17iT^{2} \)
19 \( 1 + (5.37 + 5.37i)T + 19iT^{2} \)
23 \( 1 + (-3.08 - 3.08i)T + 23iT^{2} \)
29 \( 1 + (-4.22 + 4.22i)T - 29iT^{2} \)
31 \( 1 + 8.75iT - 31T^{2} \)
37 \( 1 - 5.41iT - 37T^{2} \)
41 \( 1 - 2.54iT - 41T^{2} \)
43 \( 1 + 4.30iT - 43T^{2} \)
47 \( 1 + (-4.56 - 4.56i)T + 47iT^{2} \)
53 \( 1 - 6.07T + 53T^{2} \)
59 \( 1 + (7.33 - 7.33i)T - 59iT^{2} \)
61 \( 1 + (4.81 + 4.81i)T + 61iT^{2} \)
67 \( 1 - 14.3iT - 67T^{2} \)
71 \( 1 - 2.97T + 71T^{2} \)
73 \( 1 + (-6.87 + 6.87i)T - 73iT^{2} \)
79 \( 1 - 10.1T + 79T^{2} \)
83 \( 1 - 7.15T + 83T^{2} \)
89 \( 1 + 1.10T + 89T^{2} \)
97 \( 1 + (-7.15 + 7.15i)T - 97iT^{2} \)
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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

−11.36269718706074739676570157661, −10.60053941763559023998102420701, −9.362872763616423464997435730152, −8.495391871920910986702709153546, −7.72157550182139914889215359668, −6.22055935928601732766877996474, −5.35042616170829928262152811313, −4.39030650150776445955721284104, −2.74051281981871487175196904781, −0.60705484405449723320605318930, 2.22664393660436859106675981715, 3.58954923358416996914856367121, 4.98252592051903412828630919381, 6.22905281329739194752933001377, 6.88174340177603819553784099387, 8.156397878404055624034841118029, 9.054250331996990458814088413439, 10.48791845115481908938124940481, 10.79560728503505961638383358566, 11.91677223851451678470981920499

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