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Properties

Label 2-320-16.5-c1-0-0
Degree $2$
Conductor $320$
Sign $-0.999 + 0.00792i$
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  + (−1.37 + 1.37i)3-s + (−0.707 − 0.707i)5-s + 2.73i·7-s − 0.755i·9-s + (−4.12 − 4.12i)11-s + (−1.37 + 1.37i)13-s + 1.93·15-s − 4.94·17-s + (0.292 − 0.292i)19-s + (−3.74 − 3.74i)21-s + 1.64i·23-s + 1.00i·25-s + (−3.07 − 3.07i)27-s + (−5.67 + 5.67i)29-s − 3.95·31-s + ⋯
L(s)  = 1  + (−0.791 + 0.791i)3-s + (−0.316 − 0.316i)5-s + 1.03i·7-s − 0.251i·9-s + (−1.24 − 1.24i)11-s + (−0.382 + 0.382i)13-s + 0.500·15-s − 1.20·17-s + (0.0671 − 0.0671i)19-s + (−0.817 − 0.817i)21-s + 0.343i·23-s + 0.200i·25-s + (−0.591 − 0.591i)27-s + (−1.05 + 1.05i)29-s − 0.710·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.00792i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.00792i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(320\)    =    \(2^{6} \cdot 5\)
Sign: $-0.999 + 0.00792i$
Analytic conductor: \(2.55521\)
Root analytic conductor: \(1.59850\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{320} (241, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 320,\ (\ :1/2),\ -0.999 + 0.00792i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00123495 - 0.311581i\)
\(L(\frac12)\) \(\approx\) \(0.00123495 - 0.311581i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.37 - 1.37i)T - 3iT^{2} \)
7 \( 1 - 2.73iT - 7T^{2} \)
11 \( 1 + (4.12 + 4.12i)T + 11iT^{2} \)
13 \( 1 + (1.37 - 1.37i)T - 13iT^{2} \)
17 \( 1 + 4.94T + 17T^{2} \)
19 \( 1 + (-0.292 + 0.292i)T - 19iT^{2} \)
23 \( 1 - 1.64iT - 23T^{2} \)
29 \( 1 + (5.67 - 5.67i)T - 29iT^{2} \)
31 \( 1 + 3.95T + 31T^{2} \)
37 \( 1 + (-2.48 - 2.48i)T + 37iT^{2} \)
41 \( 1 + 8.40iT - 41T^{2} \)
43 \( 1 + (-3.22 - 3.22i)T + 43iT^{2} \)
47 \( 1 - 5.19T + 47T^{2} \)
53 \( 1 + (-7.20 - 7.20i)T + 53iT^{2} \)
59 \( 1 + (-6.41 - 6.41i)T + 59iT^{2} \)
61 \( 1 + (3.82 - 3.82i)T - 61iT^{2} \)
67 \( 1 + (5.76 - 5.76i)T - 67iT^{2} \)
71 \( 1 - 7.92iT - 71T^{2} \)
73 \( 1 - 4.36iT - 73T^{2} \)
79 \( 1 - 5.56T + 79T^{2} \)
83 \( 1 + (-0.516 + 0.516i)T - 83iT^{2} \)
89 \( 1 + 6.42iT - 89T^{2} \)
97 \( 1 + 9.44T + 97T^{2} \)
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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

−11.85090457172486860368030490616, −11.10880569942784408782952140463, −10.52606254047826371065266679797, −9.213251851446359028507877611522, −8.578927357153094101431469546829, −7.33184930548921033020934455115, −5.72876753452998933223727935987, −5.35734570135572313857770127479, −4.14391195971021712299537575939, −2.56196368929201488391168078048, 0.22913001890488435988619933548, 2.23142115404389441142057702130, 4.04879777391325185783096601789, 5.19564059469759680394095792672, 6.47845567722945697740900451464, 7.32489472996001083085889517655, 7.80860235494637578313358485990, 9.490208143971115811538914113600, 10.50693187881403278902517439572, 11.12409231912773785340054185222

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