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Properties

Label 2-320-20.3-c1-0-8
Degree $2$
Conductor $320$
Sign $0.525 + 0.850i$
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 − 2i)3-s + (2 + i)5-s + (−2 − 2i)7-s − 5i·9-s + (1 + i)13-s + (6 − 2i)15-s + (−5 + 5i)17-s + 4·19-s − 8·21-s + (−2 + 2i)23-s + (3 + 4i)25-s + (−4 − 4i)27-s − 4i·29-s − 4i·31-s + (−2 − 6i)35-s + ⋯
L(s)  = 1  + (1.15 − 1.15i)3-s + (0.894 + 0.447i)5-s + (−0.755 − 0.755i)7-s − 1.66i·9-s + (0.277 + 0.277i)13-s + (1.54 − 0.516i)15-s + (−1.21 + 1.21i)17-s + 0.917·19-s − 1.74·21-s + (−0.417 + 0.417i)23-s + (0.600 + 0.800i)25-s + (−0.769 − 0.769i)27-s − 0.742i·29-s − 0.718i·31-s + (−0.338 − 1.01i)35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.525 + 0.850i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.525 + 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(320\)    =    \(2^{6} \cdot 5\)
Sign: $0.525 + 0.850i$
Analytic conductor: \(2.55521\)
Root analytic conductor: \(1.59850\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{320} (63, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 320,\ (\ :1/2),\ 0.525 + 0.850i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.70817 - 0.952368i\)
\(L(\frac12)\) \(\approx\) \(1.70817 - 0.952368i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2 + 2i)T - 3iT^{2} \)
7 \( 1 + (2 + 2i)T + 7iT^{2} \)
11 \( 1 - 11T^{2} \)
13 \( 1 + (-1 - i)T + 13iT^{2} \)
17 \( 1 + (5 - 5i)T - 17iT^{2} \)
19 \( 1 - 4T + 19T^{2} \)
23 \( 1 + (2 - 2i)T - 23iT^{2} \)
29 \( 1 + 4iT - 29T^{2} \)
31 \( 1 + 4iT - 31T^{2} \)
37 \( 1 + (1 - i)T - 37iT^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 + (6 - 6i)T - 43iT^{2} \)
47 \( 1 + (-2 - 2i)T + 47iT^{2} \)
53 \( 1 + (-7 - 7i)T + 53iT^{2} \)
59 \( 1 - 4T + 59T^{2} \)
61 \( 1 - 4T + 61T^{2} \)
67 \( 1 + (10 + 10i)T + 67iT^{2} \)
71 \( 1 - 12iT - 71T^{2} \)
73 \( 1 + (3 + 3i)T + 73iT^{2} \)
79 \( 1 + 16T + 79T^{2} \)
83 \( 1 + (2 - 2i)T - 83iT^{2} \)
89 \( 1 - 89T^{2} \)
97 \( 1 + (3 - 3i)T - 97iT^{2} \)
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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

−11.57315681538910339168721901959, −10.34640463301166789617916303982, −9.524014131081033781687273619419, −8.623402503323118758598025292129, −7.55229769301900557870915055929, −6.74232386529797678794549029764, −6.02200135929139039453656588515, −3.91646776746271486704453338781, −2.75590623724216032251336061566, −1.59133414528139497970127095113, 2.36412307366741810259947605973, 3.28845776483147009444881669870, 4.69476694812651960367409068810, 5.61286764504178668945953674496, 6.97603214422178413439368945778, 8.648911364248645495779822676460, 8.933495585803860623297955552005, 9.779715327005286913427318400166, 10.40154355201536333201077454439, 11.76879059408955910240786623277

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