Properties

Label 2-320-80.27-c1-0-3
Degree $2$
Conductor $320$
Sign $0.811 - 0.584i$
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  + 2·3-s + (−2 + i)5-s + (3 + 3i)7-s + 9-s + (1 − i)11-s + 2i·13-s + (−4 + 2i)15-s + (1 + i)17-s + (3 − 3i)19-s + (6 + 6i)21-s + (1 − i)23-s + (3 − 4i)25-s − 4·27-s + (−7 − 7i)29-s + 2i·31-s + ⋯
L(s)  = 1  + 1.15·3-s + (−0.894 + 0.447i)5-s + (1.13 + 1.13i)7-s + 0.333·9-s + (0.301 − 0.301i)11-s + 0.554i·13-s + (−1.03 + 0.516i)15-s + (0.242 + 0.242i)17-s + (0.688 − 0.688i)19-s + (1.30 + 1.30i)21-s + (0.208 − 0.208i)23-s + (0.600 − 0.800i)25-s − 0.769·27-s + (−1.29 − 1.29i)29-s + 0.359i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.811 - 0.584i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.811 - 0.584i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(320\)    =    \(2^{6} \cdot 5\)
Sign: $0.811 - 0.584i$
Analytic conductor: \(2.55521\)
Root analytic conductor: \(1.59850\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{320} (207, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 320,\ (\ :1/2),\ 0.811 - 0.584i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.68941 + 0.545382i\)
\(L(\frac12)\) \(\approx\) \(1.68941 + 0.545382i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2 - i)T \)
good3 \( 1 - 2T + 3T^{2} \)
7 \( 1 + (-3 - 3i)T + 7iT^{2} \)
11 \( 1 + (-1 + i)T - 11iT^{2} \)
13 \( 1 - 2iT - 13T^{2} \)
17 \( 1 + (-1 - i)T + 17iT^{2} \)
19 \( 1 + (-3 + 3i)T - 19iT^{2} \)
23 \( 1 + (-1 + i)T - 23iT^{2} \)
29 \( 1 + (7 + 7i)T + 29iT^{2} \)
31 \( 1 - 2iT - 31T^{2} \)
37 \( 1 + 6iT - 37T^{2} \)
41 \( 1 - 4iT - 41T^{2} \)
43 \( 1 - 4iT - 43T^{2} \)
47 \( 1 + (-7 + 7i)T - 47iT^{2} \)
53 \( 1 + 8T + 53T^{2} \)
59 \( 1 + (3 + 3i)T + 59iT^{2} \)
61 \( 1 + (1 - i)T - 61iT^{2} \)
67 \( 1 + 4iT - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + (-3 - 3i)T + 73iT^{2} \)
79 \( 1 + 8T + 79T^{2} \)
83 \( 1 - 2T + 83T^{2} \)
89 \( 1 + 6T + 89T^{2} \)
97 \( 1 + (11 + 11i)T + 97iT^{2} \)
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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

−11.50814938353155228541648729155, −11.18527061447747119851428558245, −9.528214016195289226326838627861, −8.742467991714931292882188491978, −8.085843928995543131546044172507, −7.27628014750412082123667030193, −5.77524046318002532213843835306, −4.42050212556708163117585892987, −3.22925393380734205009349706187, −2.11640262027913904639651854572, 1.40955617353133347827143458429, 3.30341074745633033539460164973, 4.15979097804442808367545409375, 5.29173233385468064927013894560, 7.35378707532596062831670289749, 7.69993671153038351534142081142, 8.545716628297638734549363976848, 9.478182554347819723251539097819, 10.68888766525442977472543815130, 11.49078568171744158644264814125

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