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Properties

Label 2-320-80.43-c1-0-9
Degree $2$
Conductor $320$
Sign $-0.723 + 0.690i$
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  − 1.96i·3-s + (−1.72 − 1.42i)5-s + (1.60 − 1.60i)7-s − 0.851·9-s + (−0.754 + 0.754i)11-s − 5.94·13-s + (−2.79 + 3.38i)15-s + (1.95 − 1.95i)17-s + (0.780 − 0.780i)19-s + (−3.14 − 3.14i)21-s + (−4.93 − 4.93i)23-s + (0.956 + 4.90i)25-s − 4.21i·27-s + (1.44 + 1.44i)29-s + 3.60i·31-s + ⋯
L(s)  = 1  − 1.13i·3-s + (−0.771 − 0.635i)5-s + (0.605 − 0.605i)7-s − 0.283·9-s + (−0.227 + 0.227i)11-s − 1.64·13-s + (−0.720 + 0.874i)15-s + (0.474 − 0.474i)17-s + (0.179 − 0.179i)19-s + (−0.686 − 0.686i)21-s + (−1.02 − 1.02i)23-s + (0.191 + 0.981i)25-s − 0.811i·27-s + (0.268 + 0.268i)29-s + 0.648i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.723 + 0.690i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.723 + 0.690i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(320\)    =    \(2^{6} \cdot 5\)
Sign: $-0.723 + 0.690i$
Analytic conductor: \(2.55521\)
Root analytic conductor: \(1.59850\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{320} (143, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 320,\ (\ :1/2),\ -0.723 + 0.690i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.383759 - 0.958403i\)
\(L(\frac12)\) \(\approx\) \(0.383759 - 0.958403i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.72 + 1.42i)T \)
good3 \( 1 + 1.96iT - 3T^{2} \)
7 \( 1 + (-1.60 + 1.60i)T - 7iT^{2} \)
11 \( 1 + (0.754 - 0.754i)T - 11iT^{2} \)
13 \( 1 + 5.94T + 13T^{2} \)
17 \( 1 + (-1.95 + 1.95i)T - 17iT^{2} \)
19 \( 1 + (-0.780 + 0.780i)T - 19iT^{2} \)
23 \( 1 + (4.93 + 4.93i)T + 23iT^{2} \)
29 \( 1 + (-1.44 - 1.44i)T + 29iT^{2} \)
31 \( 1 - 3.60iT - 31T^{2} \)
37 \( 1 - 10.2T + 37T^{2} \)
41 \( 1 + 6.93iT - 41T^{2} \)
43 \( 1 - 9.91T + 43T^{2} \)
47 \( 1 + (-0.104 - 0.104i)T + 47iT^{2} \)
53 \( 1 + 4.03iT - 53T^{2} \)
59 \( 1 + (3.46 + 3.46i)T + 59iT^{2} \)
61 \( 1 + (-0.680 + 0.680i)T - 61iT^{2} \)
67 \( 1 - 9.04T + 67T^{2} \)
71 \( 1 - 3.64T + 71T^{2} \)
73 \( 1 + (-2.94 + 2.94i)T - 73iT^{2} \)
79 \( 1 - 10.7T + 79T^{2} \)
83 \( 1 - 4.23iT - 83T^{2} \)
89 \( 1 + 0.0426T + 89T^{2} \)
97 \( 1 + (1.91 - 1.91i)T - 97iT^{2} \)
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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

−11.60018308755685872521394392906, −10.43152623644594268952179961600, −9.332885604502736665917577437425, −7.892626875578381001693889587468, −7.69703519998723193611545649375, −6.73353987072123468287478646021, −5.13432428781327044533979468158, −4.24633466472743996965952621865, −2.36033822447009540166649040818, −0.75529512248551798654444764974, 2.57624148820347850320806200533, 3.89024606072525599246646003638, 4.79624476390644386815545474724, 5.89296416725353158317459671946, 7.47919527758215696863264603343, 8.079950785289378760132198526835, 9.489599840270622888924106788266, 10.04808075626164053276155216011, 11.06397233766942747546452757125, 11.76715197922393096437780043940

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