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Properties

Label 2-640-80.3-c1-0-11
Degree $2$
Conductor $640$
Sign $0.955 - 0.296i$
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  + 2.55·3-s + (−1.49 + 1.66i)5-s + (2.40 − 2.40i)7-s + 3.51·9-s + (2.67 + 2.67i)11-s + 2.40i·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 + 1.30·27-s + (3.95 − 3.95i)29-s + 1.65i·31-s + ⋯
L(s)  = 1  + 1.47·3-s + (−0.666 + 0.745i)5-s + (0.908 − 0.908i)7-s + 1.17·9-s + (0.807 + 0.807i)11-s + 0.666i·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 + 0.250·27-s + (0.734 − 0.734i)29-s + 0.297i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.955 - 0.296i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.955 - 0.296i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(640\)    =    \(2^{7} \cdot 5\)
Sign: $0.955 - 0.296i$
Analytic conductor: \(5.11042\)
Root analytic conductor: \(2.26062\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{640} (223, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 640,\ (\ :1/2),\ 0.955 - 0.296i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.38485 + 0.361161i\)
\(L(\frac12)\) \(\approx\) \(2.38485 + 0.361161i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.49 - 1.66i)T \)
good3 \( 1 - 2.55T + 3T^{2} \)
7 \( 1 + (-2.40 + 2.40i)T - 7iT^{2} \)
11 \( 1 + (-2.67 - 2.67i)T + 11iT^{2} \)
13 \( 1 - 2.40iT - 13T^{2} \)
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 \( 1 + 2.53iT - 37T^{2} \)
41 \( 1 + 1.70iT - 41T^{2} \)
43 \( 1 - 3.84iT - 43T^{2} \)
47 \( 1 + (-2.15 - 2.15i)T + 47iT^{2} \)
53 \( 1 - 1.29T + 53T^{2} \)
59 \( 1 + (5.29 - 5.29i)T - 59iT^{2} \)
61 \( 1 + (10.2 + 10.2i)T + 61iT^{2} \)
67 \( 1 + 10.6iT - 67T^{2} \)
71 \( 1 - 2.27T + 71T^{2} \)
73 \( 1 + (9.99 - 9.99i)T - 73iT^{2} \)
79 \( 1 + 8.70T + 79T^{2} \)
83 \( 1 + 11.1T + 83T^{2} \)
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

−10.60893205167862429202324290119, −9.596455488725606830867186553780, −8.813014673437654859262121690044, −7.899326661193758393945321295454, −7.33095773179906608127027277327, −6.57764477321297680727064639755, −4.49788761682531750444072903876, −4.05487176563477930192333770680, −2.88133732870256667933750247637, −1.68971870642796791159777000190, 1.44553475557915271226638302288, 2.78853848278966741891807068471, 3.78673955456166920349123599087, 4.80342037884006332479341604962, 5.93756809959369505069649729761, 7.40129045237825640912005502082, 8.328711041811600840539951552679, 8.591556170312290140341065076485, 9.183926478452157013417950212062, 10.46352480442267380803396902577

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