Properties

Label 2-640-8.5-c1-0-9
Degree $2$
Conductor $640$
Sign $0.707 + 0.707i$
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  + 1.41i·3-s i·5-s − 4.24·7-s + 0.999·9-s − 5.65i·11-s − 2i·13-s + 1.41·15-s + 6·17-s − 2.82i·19-s − 6i·21-s + 7.07·23-s − 25-s + 5.65i·27-s − 4i·29-s − 2.82·31-s + ⋯
L(s)  = 1  + 0.816i·3-s − 0.447i·5-s − 1.60·7-s + 0.333·9-s − 1.70i·11-s − 0.554i·13-s + 0.365·15-s + 1.45·17-s − 0.648i·19-s − 1.30i·21-s + 1.47·23-s − 0.200·25-s + 1.08i·27-s − 0.742i·29-s − 0.508·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(640\)    =    \(2^{7} \cdot 5\)
Sign: $0.707 + 0.707i$
Analytic conductor: \(5.11042\)
Root analytic conductor: \(2.26062\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{640} (321, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 640,\ (\ :1/2),\ 0.707 + 0.707i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.09696 - 0.454375i\)
\(L(\frac12)\) \(\approx\) \(1.09696 - 0.454375i\)
\(L(\frac{3}{2})\) 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 \( 1 + iT \)
good3 \( 1 - 1.41iT - 3T^{2} \)
7 \( 1 + 4.24T + 7T^{2} \)
11 \( 1 + 5.65iT - 11T^{2} \)
13 \( 1 + 2iT - 13T^{2} \)
17 \( 1 - 6T + 17T^{2} \)
19 \( 1 + 2.82iT - 19T^{2} \)
23 \( 1 - 7.07T + 23T^{2} \)
29 \( 1 + 4iT - 29T^{2} \)
31 \( 1 + 2.82T + 31T^{2} \)
37 \( 1 + 2iT - 37T^{2} \)
41 \( 1 + 8T + 41T^{2} \)
43 \( 1 - 1.41iT - 43T^{2} \)
47 \( 1 + 1.41T + 47T^{2} \)
53 \( 1 - 2iT - 53T^{2} \)
59 \( 1 + 2.82iT - 59T^{2} \)
61 \( 1 + 14iT - 61T^{2} \)
67 \( 1 + 4.24iT - 67T^{2} \)
71 \( 1 - 2.82T + 71T^{2} \)
73 \( 1 + 6T + 73T^{2} \)
79 \( 1 - 16.9T + 79T^{2} \)
83 \( 1 + 12.7iT - 83T^{2} \)
89 \( 1 + 6T + 89T^{2} \)
97 \( 1 - 10T + 97T^{2} \)
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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

−10.36926413967852164743846716464, −9.547626964815363223420711590115, −9.039040700517945874320302359160, −7.963424228184603107423239082070, −6.77848204638477211022921944776, −5.82232411586035505478478461304, −5.01126945023178015261143991850, −3.51306791888257889842289374559, −3.19568186379880682856062887964, −0.69360634084577674764769493279, 1.51534072924660585125836413196, 2.86764407793390778683761792912, 3.93747824915481641699339942293, 5.34612939172440329244279706201, 6.64955722366092072653421203520, 6.95263691217372191344450813372, 7.70348213657919961821157465824, 9.142449270544224216333849997726, 9.924022877947080583335173486520, 10.35400679362197395245451932633

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