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Properties

Label 2-320-20.3-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: $\chi_{320} (63, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 320,\ (\ :1/2),\ 0.0898 - 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.694610 + 0.634795i\)
\(L(\frac12)\) \(\approx\) \(0.694610 + 0.634795i\)
\(L(\frac{3}{2})\) 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

−11.90610077894629688601717844975, −11.03811215650973180610571614677, −9.890225631806539205448181678487, −9.263756994117660218074196810189, −8.005957393011283377556188545346, −7.21272644162031528486031263166, −5.50630495973484261816459782947, −4.92567796347191702071413721364, −3.95375881348507857996110734004, −1.81572903787505839753233493118, 0.76743892918537886287415781356, 2.96026386145765863367697844051, 4.08707047708737683569632907038, 5.93547372723224016811702573831, 6.29867084661856279217229405749, 7.64506178013416516018906516900, 8.232194008465131626483432211018, 9.704867874929865178424189540651, 10.84688550840180266457463482690, 11.37640160458337348389505031560

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