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Properties

Label 2-640-40.19-c2-0-20
Degree $2$
Conductor $640$
Sign $0.999 + 0.00738i$
Analytic cond. $17.4387$
Root an. cond. $4.17597$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.20i·3-s + (3.50 + 3.56i)5-s + 0.620·7-s + 4.12·9-s + 17.6·11-s − 17.9·13-s + (7.86 − 7.74i)15-s + 10.9i·17-s + 2.17·19-s − 1.36i·21-s + 13.8·23-s + (−0.369 + 24.9i)25-s − 28.9i·27-s + 14.7i·29-s + 55.2i·31-s + ⋯
L(s)  = 1  − 0.736i·3-s + (0.701 + 0.712i)5-s + 0.0885·7-s + 0.458·9-s + 1.60·11-s − 1.38·13-s + (0.524 − 0.516i)15-s + 0.644i·17-s + 0.114·19-s − 0.0652i·21-s + 0.603·23-s + (−0.0147 + 0.999i)25-s − 1.07i·27-s + 0.508i·29-s + 1.78i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00738i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.999 + 0.00738i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(640\)    =    \(2^{7} \cdot 5\)
Sign: $0.999 + 0.00738i$
Analytic conductor: \(17.4387\)
Root analytic conductor: \(4.17597\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{640} (319, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 640,\ (\ :1),\ 0.999 + 0.00738i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.313828489\)
\(L(\frac12)\) \(\approx\) \(2.313828489\)
\(L(2)\) 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 \( 1 + (-3.50 - 3.56i)T \)
good3 \( 1 + 2.20iT - 9T^{2} \)
7 \( 1 - 0.620T + 49T^{2} \)
11 \( 1 - 17.6T + 121T^{2} \)
13 \( 1 + 17.9T + 169T^{2} \)
17 \( 1 - 10.9iT - 289T^{2} \)
19 \( 1 - 2.17T + 361T^{2} \)
23 \( 1 - 13.8T + 529T^{2} \)
29 \( 1 - 14.7iT - 841T^{2} \)
31 \( 1 - 55.2iT - 961T^{2} \)
37 \( 1 - 7.01T + 1.36e3T^{2} \)
41 \( 1 - 48.3T + 1.68e3T^{2} \)
43 \( 1 + 58.7iT - 1.84e3T^{2} \)
47 \( 1 - 54.1T + 2.20e3T^{2} \)
53 \( 1 - 3.94T + 2.80e3T^{2} \)
59 \( 1 - 72.8T + 3.48e3T^{2} \)
61 \( 1 + 79.3iT - 3.72e3T^{2} \)
67 \( 1 + 26.7iT - 4.48e3T^{2} \)
71 \( 1 + 46.4iT - 5.04e3T^{2} \)
73 \( 1 - 95.1iT - 5.32e3T^{2} \)
79 \( 1 - 24.2iT - 6.24e3T^{2} \)
83 \( 1 + 107. iT - 6.88e3T^{2} \)
89 \( 1 - 44.2T + 7.92e3T^{2} \)
97 \( 1 + 123. iT - 9.40e3T^{2} \)
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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

−10.29527331676073486288976364346, −9.562044422150437285405675070224, −8.712250049391622272838790858628, −7.32454315565537085009285930007, −6.93749406061409983652552900765, −6.12882725769243901254627300998, −4.90250254527277593828723922242, −3.61310507487687778082192332291, −2.26635746492065260745636498976, −1.27801771441844193524773276360, 1.03975880402430727176604161193, 2.45467468175257864789508865219, 4.08845086406366163669468291285, 4.66605422909714916379232220431, 5.68852921093323948728079736999, 6.76573990868619225906804177097, 7.73595776132286795433726568770, 9.103386777317542379438076866911, 9.486830872185576600999443288717, 10.01307028141117624143461083380

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