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Properties

Label 2-320-80.37-c2-0-17
Degree $2$
Conductor $320$
Sign $-0.0228 + 0.999i$
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  − 1.50i·3-s + (3.69 − 3.37i)5-s + (1.28 − 1.28i)7-s + 6.72·9-s + (−13.2 − 13.2i)11-s + 0.642i·13-s + (−5.08 − 5.56i)15-s + (5.68 + 5.68i)17-s + (15.5 + 15.5i)19-s + (−1.93 − 1.93i)21-s + (−8.84 − 8.84i)23-s + (2.26 − 24.8i)25-s − 23.7i·27-s + (−28.0 − 28.0i)29-s − 8.45·31-s + ⋯
L(s)  = 1  − 0.502i·3-s + (0.738 − 0.674i)5-s + (0.183 − 0.183i)7-s + 0.747·9-s + (−1.20 − 1.20i)11-s + 0.0494i·13-s + (−0.338 − 0.370i)15-s + (0.334 + 0.334i)17-s + (0.819 + 0.819i)19-s + (−0.0919 − 0.0919i)21-s + (−0.384 − 0.384i)23-s + (0.0907 − 0.995i)25-s − 0.877i·27-s + (−0.967 − 0.967i)29-s − 0.272·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0228 + 0.999i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0228 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(320\)    =    \(2^{6} \cdot 5\)
Sign: $-0.0228 + 0.999i$
Analytic conductor: \(8.71936\)
Root analytic conductor: \(2.95285\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{320} (177, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 320,\ (\ :1),\ -0.0228 + 0.999i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.25976 - 1.28889i\)
\(L(\frac12)\) \(\approx\) \(1.25976 - 1.28889i\)
\(L(2)\) 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 \( 1 + (-3.69 + 3.37i)T \)
good3 \( 1 + 1.50iT - 9T^{2} \)
7 \( 1 + (-1.28 + 1.28i)T - 49iT^{2} \)
11 \( 1 + (13.2 + 13.2i)T + 121iT^{2} \)
13 \( 1 - 0.642iT - 169T^{2} \)
17 \( 1 + (-5.68 - 5.68i)T + 289iT^{2} \)
19 \( 1 + (-15.5 - 15.5i)T + 361iT^{2} \)
23 \( 1 + (8.84 + 8.84i)T + 529iT^{2} \)
29 \( 1 + (28.0 + 28.0i)T + 841iT^{2} \)
31 \( 1 + 8.45T + 961T^{2} \)
37 \( 1 + 71.1iT - 1.36e3T^{2} \)
41 \( 1 - 26.7iT - 1.68e3T^{2} \)
43 \( 1 - 53.5T + 1.84e3T^{2} \)
47 \( 1 + (-2.27 - 2.27i)T + 2.20e3iT^{2} \)
53 \( 1 - 4.55T + 2.80e3T^{2} \)
59 \( 1 + (-38.4 + 38.4i)T - 3.48e3iT^{2} \)
61 \( 1 + (47.3 - 47.3i)T - 3.72e3iT^{2} \)
67 \( 1 - 84.1T + 4.48e3T^{2} \)
71 \( 1 - 48.7iT - 5.04e3T^{2} \)
73 \( 1 + (11.1 + 11.1i)T + 5.32e3iT^{2} \)
79 \( 1 - 49.3iT - 6.24e3T^{2} \)
83 \( 1 - 134. iT - 6.88e3T^{2} \)
89 \( 1 + 74.7T + 7.92e3T^{2} \)
97 \( 1 + (-112. - 112. i)T + 9.40e3iT^{2} \)
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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

−11.11080279725988962363829704796, −10.22351982333047726633332120193, −9.360706501195930432714358940255, −8.155157758007832955146159446947, −7.54905942400967230474004189943, −6.05131183260666129055705059040, −5.41611728768772437506631270550, −3.97920385514052676412251915572, −2.28020630399437502544684039087, −0.894947576904974912847271216116, 1.90520374960999877694604962904, 3.19125461821179011863438535898, 4.75872648933639164150799062963, 5.50731603147719904050325677823, 7.00627883353358468798199752347, 7.57947377483744435189501012987, 9.166238749405445094905291371838, 9.939804619701208161998729583720, 10.45527105273665031865309730743, 11.48892041572485244914609626055

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