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Properties

Label 2-640-5.4-c1-0-4
Degree $2$
Conductor $640$
Sign $0.241 - 0.970i$
Analytic cond. $5.11042$
Root an. cond. $2.26062$
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  + 1.70i·3-s + (−2.17 − 0.539i)5-s − 2.63i·7-s + 0.0783·9-s + 5.41·11-s + 6.34i·13-s + (0.921 − 3.70i)15-s + 3.41i·17-s − 3.26·19-s + 4.49·21-s + 1.36i·23-s + (4.41 + 2.34i)25-s + 5.26i·27-s + 2·29-s + 4.68·31-s + ⋯
L(s)  = 1  + 0.986i·3-s + (−0.970 − 0.241i)5-s − 0.994i·7-s + 0.0261·9-s + 1.63·11-s + 1.75i·13-s + (0.237 − 0.957i)15-s + 0.829i·17-s − 0.748·19-s + 0.981·21-s + 0.285i·23-s + (0.883 + 0.468i)25-s + 1.01i·27-s + 0.371·29-s + 0.840·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.241 - 0.970i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.241 - 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(640\)    =    \(2^{7} \cdot 5\)
Sign: $0.241 - 0.970i$
Analytic conductor: \(5.11042\)
Root analytic conductor: \(2.26062\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{640} (129, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 640,\ (\ :1/2),\ 0.241 - 0.970i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.04013 + 0.813323i\)
\(L(\frac12)\) \(\approx\) \(1.04013 + 0.813323i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.17 + 0.539i)T \)
good3 \( 1 - 1.70iT - 3T^{2} \)
7 \( 1 + 2.63iT - 7T^{2} \)
11 \( 1 - 5.41T + 11T^{2} \)
13 \( 1 - 6.34iT - 13T^{2} \)
17 \( 1 - 3.41iT - 17T^{2} \)
19 \( 1 + 3.26T + 19T^{2} \)
23 \( 1 - 1.36iT - 23T^{2} \)
29 \( 1 - 2T + 29T^{2} \)
31 \( 1 - 4.68T + 31T^{2} \)
37 \( 1 - 5.75iT - 37T^{2} \)
41 \( 1 + 7.75T + 41T^{2} \)
43 \( 1 + 4.44iT - 43T^{2} \)
47 \( 1 + 4.78iT - 47T^{2} \)
53 \( 1 - 1.65iT - 53T^{2} \)
59 \( 1 - 3.26T + 59T^{2} \)
61 \( 1 - 2.49T + 61T^{2} \)
67 \( 1 - 7.86iT - 67T^{2} \)
71 \( 1 - 6.15T + 71T^{2} \)
73 \( 1 - 13.5iT - 73T^{2} \)
79 \( 1 - 12.6T + 79T^{2} \)
83 \( 1 + 14.9iT - 83T^{2} \)
89 \( 1 - 8.52T + 89T^{2} \)
97 \( 1 + 4.58iT - 97T^{2} \)
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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

−10.70279657720682617389374531513, −9.927208761517127981215231072036, −9.041747066856190155060571414042, −8.381009967670667420616703259120, −7.00745982763725617832567018472, −6.58624150239615852890033729991, −4.76958914249168510742625634764, −4.00376065829358913741521227302, −3.80636356463802694250025123771, −1.43111198691620754670780777606, 0.837512602725564302670678358522, 2.45357920631419073621964914111, 3.59224656910736929562405462684, 4.85867903245936313465846955545, 6.18040911176930181377705054361, 6.80260939939804957434220099178, 7.80829434976398134761504548372, 8.409792739317479390981251361498, 9.366716494333968171501735006047, 10.52600522759019618495167719152

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