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Properties

Label 2-320-16.13-c1-0-7
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: $\chi_{320} (81, \cdot )$
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

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

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