Properties

Label 2-640-80.43-c1-0-11
Degree $2$
Conductor $640$
Sign $0.962 - 0.270i$
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  + 2.55i·3-s + (−1.66 − 1.49i)5-s + (2.40 − 2.40i)7-s − 3.51·9-s + (2.67 − 2.67i)11-s + 2.40·13-s + (3.80 − 4.25i)15-s + (−0.0750 + 0.0750i)17-s + (2.67 − 2.67i)19-s + (6.13 + 6.13i)21-s + (2.12 + 2.12i)23-s + (0.553 + 4.96i)25-s − 1.30i·27-s + (−3.95 − 3.95i)29-s − 1.65i·31-s + ⋯
L(s)  = 1  + 1.47i·3-s + (−0.745 − 0.666i)5-s + (0.908 − 0.908i)7-s − 1.17·9-s + (0.807 − 0.807i)11-s + 0.666·13-s + (0.982 − 1.09i)15-s + (−0.0182 + 0.0182i)17-s + (0.613 − 0.613i)19-s + (1.33 + 1.33i)21-s + (0.442 + 0.442i)23-s + (0.110 + 0.993i)25-s − 0.250i·27-s + (−0.734 − 0.734i)29-s − 0.297i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.962 - 0.270i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.962 - 0.270i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(640\)    =    \(2^{7} \cdot 5\)
Sign: $0.962 - 0.270i$
Analytic conductor: \(5.11042\)
Root analytic conductor: \(2.26062\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{640} (543, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 640,\ (\ :1/2),\ 0.962 - 0.270i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.50586 + 0.207579i\)
\(L(\frac12)\) \(\approx\) \(1.50586 + 0.207579i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.66 + 1.49i)T \)
good3 \( 1 - 2.55iT - 3T^{2} \)
7 \( 1 + (-2.40 + 2.40i)T - 7iT^{2} \)
11 \( 1 + (-2.67 + 2.67i)T - 11iT^{2} \)
13 \( 1 - 2.40T + 13T^{2} \)
17 \( 1 + (0.0750 - 0.0750i)T - 17iT^{2} \)
19 \( 1 + (-2.67 + 2.67i)T - 19iT^{2} \)
23 \( 1 + (-2.12 - 2.12i)T + 23iT^{2} \)
29 \( 1 + (3.95 + 3.95i)T + 29iT^{2} \)
31 \( 1 + 1.65iT - 31T^{2} \)
37 \( 1 - 2.53T + 37T^{2} \)
41 \( 1 - 1.70iT - 41T^{2} \)
43 \( 1 - 3.84T + 43T^{2} \)
47 \( 1 + (2.15 + 2.15i)T + 47iT^{2} \)
53 \( 1 - 1.29iT - 53T^{2} \)
59 \( 1 + (-5.29 - 5.29i)T + 59iT^{2} \)
61 \( 1 + (10.2 - 10.2i)T - 61iT^{2} \)
67 \( 1 - 10.6T + 67T^{2} \)
71 \( 1 - 2.27T + 71T^{2} \)
73 \( 1 + (-9.99 + 9.99i)T - 73iT^{2} \)
79 \( 1 - 8.70T + 79T^{2} \)
83 \( 1 + 11.1iT - 83T^{2} \)
89 \( 1 + 15.6T + 89T^{2} \)
97 \( 1 + (-5.00 + 5.00i)T - 97iT^{2} \)
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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

−10.87385935774720556363260230830, −9.635021222379430452452509577606, −8.993741582037723180800857315072, −8.190723598596661583050310537646, −7.27533479961962902281609579741, −5.77456985837865220742910415786, −4.78150773232613036576346366435, −4.09856878303290314698456388891, −3.43728526317631697520324446505, −1.03425308859178846381327023736, 1.37045999465159423991235663125, 2.40573070994459209842364345645, 3.78542951126223889608091662601, 5.18810771949921591586015438632, 6.35877503817724209726090299773, 7.00876090477643933941621154241, 7.86415104945598899927045795613, 8.433476815412728146384046368465, 9.509049250186513597595116581424, 10.97450676243046113530436192685

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