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Properties

Label 2-320-80.43-c1-0-2
Degree $2$
Conductor $320$
Sign $-0.607 - 0.794i$
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  + 2.55i·3-s + (1.66 + 1.49i)5-s + (−2.40 + 2.40i)7-s − 3.51·9-s + (2.67 − 2.67i)11-s − 2.40·13-s + (−3.80 + 4.25i)15-s + (−0.0750 + 0.0750i)17-s + (2.67 − 2.67i)19-s + (−6.13 − 6.13i)21-s + (−2.12 − 2.12i)23-s + (0.553 + 4.96i)25-s − 1.30i·27-s + (3.95 + 3.95i)29-s + 1.65i·31-s + ⋯
L(s)  = 1  + 1.47i·3-s + (0.745 + 0.666i)5-s + (−0.908 + 0.908i)7-s − 1.17·9-s + (0.807 − 0.807i)11-s − 0.666·13-s + (−0.982 + 1.09i)15-s + (−0.0182 + 0.0182i)17-s + (0.613 − 0.613i)19-s + (−1.33 − 1.33i)21-s + (−0.442 − 0.442i)23-s + (0.110 + 0.993i)25-s − 0.250i·27-s + (0.734 + 0.734i)29-s + 0.297i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.607 - 0.794i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.607 - 0.794i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(320\)    =    \(2^{6} \cdot 5\)
Sign: $-0.607 - 0.794i$
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: \((2,\ 320,\ (\ :1/2),\ -0.607 - 0.794i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.577799 + 1.16997i\)
\(L(\frac12)\) \(\approx\) \(0.577799 + 1.16997i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.66 - 1.49i)T \)
good3 \( 1 - 2.55iT - 3T^{2} \)
7 \( 1 + (2.40 - 2.40i)T - 7iT^{2} \)
11 \( 1 + (-2.67 + 2.67i)T - 11iT^{2} \)
13 \( 1 + 2.40T + 13T^{2} \)
17 \( 1 + (0.0750 - 0.0750i)T - 17iT^{2} \)
19 \( 1 + (-2.67 + 2.67i)T - 19iT^{2} \)
23 \( 1 + (2.12 + 2.12i)T + 23iT^{2} \)
29 \( 1 + (-3.95 - 3.95i)T + 29iT^{2} \)
31 \( 1 - 1.65iT - 31T^{2} \)
37 \( 1 + 2.53T + 37T^{2} \)
41 \( 1 - 1.70iT - 41T^{2} \)
43 \( 1 - 3.84T + 43T^{2} \)
47 \( 1 + (-2.15 - 2.15i)T + 47iT^{2} \)
53 \( 1 + 1.29iT - 53T^{2} \)
59 \( 1 + (-5.29 - 5.29i)T + 59iT^{2} \)
61 \( 1 + (-10.2 + 10.2i)T - 61iT^{2} \)
67 \( 1 - 10.6T + 67T^{2} \)
71 \( 1 + 2.27T + 71T^{2} \)
73 \( 1 + (-9.99 + 9.99i)T - 73iT^{2} \)
79 \( 1 + 8.70T + 79T^{2} \)
83 \( 1 + 11.1iT - 83T^{2} \)
89 \( 1 + 15.6T + 89T^{2} \)
97 \( 1 + (-5.00 + 5.00i)T - 97iT^{2} \)
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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

−11.73863001032408233499799303834, −10.82396751723072159421183303888, −9.961018565731437727421932482739, −9.383683633647795726951423529460, −8.697049808209885506073752156005, −6.86043831585314704329718088222, −5.94753502465018169414960998034, −5.02441893598399126031031692138, −3.56819497696460328062040057633, −2.70258427261354711225896961395, 0.994832385215482406732278965950, 2.26350065682293930208075756198, 4.06608006368123473954563074790, 5.61057627241445552803023196269, 6.66221024447933051353177646223, 7.22152638808664251395514813549, 8.288419328891452839836589744854, 9.619010193339161988954572292402, 10.04808228044235269696129414199, 11.72758811241248021262919531211

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