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Properties

Label 2-320-20.7-c1-0-4
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

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

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

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