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Properties

Label 2-320-20.7-c1-0-1
Degree $2$
Conductor $320$
Sign $-0.0898 - 0.995i$
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 + i)3-s + (−1 + 2i)5-s + (−1 + i)7-s − i·9-s + 6i·11-s + (1 − i)13-s + (−3 + i)15-s + (1 + i)17-s − 4·19-s − 2·21-s + (5 + 5i)23-s + (−3 − 4i)25-s + (4 − 4i)27-s − 8i·29-s + 2i·31-s + ⋯
L(s)  = 1  + (0.577 + 0.577i)3-s + (−0.447 + 0.894i)5-s + (−0.377 + 0.377i)7-s − 0.333i·9-s + 1.80i·11-s + (0.277 − 0.277i)13-s + (−0.774 + 0.258i)15-s + (0.242 + 0.242i)17-s − 0.917·19-s − 0.436·21-s + (1.04 + 1.04i)23-s + (−0.600 − 0.800i)25-s + (0.769 − 0.769i)27-s − 1.48i·29-s + 0.359i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0898 - 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0898 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(320\)    =    \(2^{6} \cdot 5\)
Sign: $-0.0898 - 0.995i$
Analytic conductor: \(2.55521\)
Root analytic conductor: \(1.59850\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{320} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 320,\ (\ :1/2),\ -0.0898 - 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.906654 + 0.992085i\)
\(L(\frac12)\) \(\approx\) \(0.906654 + 0.992085i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1 - 2i)T \)
good3 \( 1 + (-1 - i)T + 3iT^{2} \)
7 \( 1 + (1 - i)T - 7iT^{2} \)
11 \( 1 - 6iT - 11T^{2} \)
13 \( 1 + (-1 + i)T - 13iT^{2} \)
17 \( 1 + (-1 - i)T + 17iT^{2} \)
19 \( 1 + 4T + 19T^{2} \)
23 \( 1 + (-5 - 5i)T + 23iT^{2} \)
29 \( 1 + 8iT - 29T^{2} \)
31 \( 1 - 2iT - 31T^{2} \)
37 \( 1 + (-5 - 5i)T + 37iT^{2} \)
41 \( 1 - 6T + 41T^{2} \)
43 \( 1 + (3 + 3i)T + 43iT^{2} \)
47 \( 1 + (-7 + 7i)T - 47iT^{2} \)
53 \( 1 + (-1 + i)T - 53iT^{2} \)
59 \( 1 + 4T + 59T^{2} \)
61 \( 1 + 2T + 61T^{2} \)
67 \( 1 + (-7 + 7i)T - 67iT^{2} \)
71 \( 1 + 6iT - 71T^{2} \)
73 \( 1 + (-9 + 9i)T - 73iT^{2} \)
79 \( 1 + 8T + 79T^{2} \)
83 \( 1 + (-5 - 5i)T + 83iT^{2} \)
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

−11.92828920887095361776041972633, −10.78840241851468527504478962394, −9.897535205114377250911723937345, −9.310491780430194098450854394124, −8.083294707773213364670804230382, −7.11186202054915336867025013045, −6.15927039959388504878409697851, −4.53537320768920545153705018769, −3.56947203076070380168261943174, −2.42352590918712853466567242207, 0.965135808481614986405658292943, 2.83914565713673064494934908551, 4.08831241527517536506397399518, 5.41066169742392480345008753508, 6.63788300244926264859356528384, 7.76114321371533072991784338575, 8.578585476202832637249814237153, 9.086184361996998785091766103786, 10.69693638849651829150433791638, 11.28628376592839010459808379064

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