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Properties

Label 2-640-128.109-c1-0-63
Degree $2$
Conductor $640$
Sign $-0.976 + 0.215i$
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  + (0.823 − 1.15i)2-s + (1.18 + 0.635i)3-s + (−0.645 − 1.89i)4-s + (−0.634 + 0.773i)5-s + (1.70 − 0.843i)6-s + (−4.07 − 2.72i)7-s + (−2.70 − 0.815i)8-s + (−0.658 − 0.985i)9-s + (0.366 + 1.36i)10-s + (−5.20 − 1.57i)11-s + (0.435 − 2.65i)12-s + (3.21 − 2.63i)13-s + (−6.48 + 2.44i)14-s + (−1.24 + 0.515i)15-s + (−3.16 + 2.44i)16-s + (3.95 + 1.64i)17-s + ⋯
L(s)  = 1  + (0.581 − 0.813i)2-s + (0.686 + 0.366i)3-s + (−0.322 − 0.946i)4-s + (−0.283 + 0.345i)5-s + (0.697 − 0.344i)6-s + (−1.53 − 1.02i)7-s + (−0.957 − 0.288i)8-s + (−0.219 − 0.328i)9-s + (0.116 + 0.431i)10-s + (−1.56 − 0.476i)11-s + (0.125 − 0.767i)12-s + (0.891 − 0.731i)13-s + (−1.73 + 0.653i)14-s + (−0.321 + 0.133i)15-s + (−0.791 + 0.610i)16-s + (0.960 + 0.397i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.976 + 0.215i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.976 + 0.215i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(640\)    =    \(2^{7} \cdot 5\)
Sign: $-0.976 + 0.215i$
Analytic conductor: \(5.11042\)
Root analytic conductor: \(2.26062\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{640} (621, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 640,\ (\ :1/2),\ -0.976 + 0.215i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.133329 - 1.22255i\)
\(L(\frac12)\) \(\approx\) \(0.133329 - 1.22255i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.823 + 1.15i)T \)
5 \( 1 + (0.634 - 0.773i)T \)
good3 \( 1 + (-1.18 - 0.635i)T + (1.66 + 2.49i)T^{2} \)
7 \( 1 + (4.07 + 2.72i)T + (2.67 + 6.46i)T^{2} \)
11 \( 1 + (5.20 + 1.57i)T + (9.14 + 6.11i)T^{2} \)
13 \( 1 + (-3.21 + 2.63i)T + (2.53 - 12.7i)T^{2} \)
17 \( 1 + (-3.95 - 1.64i)T + (12.0 + 12.0i)T^{2} \)
19 \( 1 + (0.725 - 7.36i)T + (-18.6 - 3.70i)T^{2} \)
23 \( 1 + (0.125 + 0.0249i)T + (21.2 + 8.80i)T^{2} \)
29 \( 1 + (0.673 + 2.21i)T + (-24.1 + 16.1i)T^{2} \)
31 \( 1 + (-6.08 + 6.08i)T - 31iT^{2} \)
37 \( 1 + (2.21 - 0.217i)T + (36.2 - 7.21i)T^{2} \)
41 \( 1 + (-2.29 + 11.5i)T + (-37.8 - 15.6i)T^{2} \)
43 \( 1 + (1.39 - 0.745i)T + (23.8 - 35.7i)T^{2} \)
47 \( 1 + (-2.64 + 6.39i)T + (-33.2 - 33.2i)T^{2} \)
53 \( 1 + (-1.26 + 4.17i)T + (-44.0 - 29.4i)T^{2} \)
59 \( 1 + (1.98 + 1.62i)T + (11.5 + 57.8i)T^{2} \)
61 \( 1 + (0.230 - 0.431i)T + (-33.8 - 50.7i)T^{2} \)
67 \( 1 + (4.49 - 8.40i)T + (-37.2 - 55.7i)T^{2} \)
71 \( 1 + (-3.72 + 5.57i)T + (-27.1 - 65.5i)T^{2} \)
73 \( 1 + (-3.99 + 2.67i)T + (27.9 - 67.4i)T^{2} \)
79 \( 1 + (2.35 + 5.69i)T + (-55.8 + 55.8i)T^{2} \)
83 \( 1 + (6.14 + 0.605i)T + (81.4 + 16.1i)T^{2} \)
89 \( 1 + (2.48 - 0.494i)T + (82.2 - 34.0i)T^{2} \)
97 \( 1 + (1.35 - 1.35i)T - 97iT^{2} \)
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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

−10.28010006349828799301153543596, −9.777040220500500792859341644845, −8.503806691494550901997262566078, −7.68478297833541464014902334665, −6.22614288097513270996430455462, −5.67447187367798370110536679174, −3.82468646294150955989788406262, −3.57380600132465621610164205031, −2.70384711750067323790100521499, −0.48101149615792726524400469515, 2.70263923327728635300213512438, 3.10061309992106238358794899013, 4.72136070917921146273347353691, 5.55895532031661480005499963756, 6.57063464432237178493167061170, 7.39832742693370488501814966539, 8.343927097388516228014576851229, 8.922340875272599660743467386861, 9.736771426986255949513466332547, 11.14858746639544451352611397790

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