Properties

Label 2-320-5.4-c1-0-1
Degree $2$
Conductor $320$
Sign $0.447 - 0.894i$
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 + 2i)5-s + 3·9-s + 4i·13-s + 8i·17-s + (−3 − 4i)25-s + 10·29-s − 12i·37-s − 10·41-s + (−3 + 6i)45-s + 7·49-s − 4i·53-s − 10·61-s + (−8 − 4i)65-s − 16i·73-s + 9·81-s + ⋯
L(s)  = 1  + (−0.447 + 0.894i)5-s + 9-s + 1.10i·13-s + 1.94i·17-s + (−0.600 − 0.800i)25-s + 1.85·29-s − 1.97i·37-s − 1.56·41-s + (−0.447 + 0.894i)45-s + 49-s − 0.549i·53-s − 1.28·61-s + (−0.992 − 0.496i)65-s − 1.87i·73-s + 81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(320\)    =    \(2^{6} \cdot 5\)
Sign: $0.447 - 0.894i$
Analytic conductor: \(2.55521\)
Root analytic conductor: \(1.59850\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{320} (129, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 320,\ (\ :1/2),\ 0.447 - 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.05471 + 0.651851i\)
\(L(\frac12)\) \(\approx\) \(1.05471 + 0.651851i\)
\(L(\frac{3}{2})\) 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 \( 1 - 3T^{2} \)
7 \( 1 - 7T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 - 4iT - 13T^{2} \)
17 \( 1 - 8iT - 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 - 10T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + 12iT - 37T^{2} \)
41 \( 1 + 10T + 41T^{2} \)
43 \( 1 - 43T^{2} \)
47 \( 1 - 47T^{2} \)
53 \( 1 + 4iT - 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 + 10T + 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 16iT - 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 - 83T^{2} \)
89 \( 1 - 10T + 89T^{2} \)
97 \( 1 + 8iT - 97T^{2} \)
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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

−11.84408354461813868935945368764, −10.66335378443347160423268801462, −10.24746998899160334246831870530, −8.964854998704914846361387367472, −7.900066297661956208511967360682, −6.91429802711173902600523257175, −6.20057686725751914499571541580, −4.47609537034312021662712926526, −3.62020807869565141328442857275, −1.92886127396355264597428082082, 0.989944096178904175251613966692, 3.03186958667187543938212924882, 4.52579293529095231292682208738, 5.20800393949219296852719967035, 6.75582600729857951182517827055, 7.71380422747132694341431362028, 8.583811000670341614167614238718, 9.662897648802038270966265082053, 10.39603958736884734475203446268, 11.74444734168646895307638647682

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