Properties

Label 2-640-80.27-c1-0-19
Degree $2$
Conductor $640$
Sign $-0.990 + 0.137i$
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  + 1.39·3-s + (−2.17 − 0.535i)5-s + (−2.13 − 2.13i)7-s − 1.05·9-s + (−2.17 + 2.17i)11-s + 1.54i·13-s + (−3.02 − 0.745i)15-s + (−3.86 − 3.86i)17-s + (−0.0136 + 0.0136i)19-s + (−2.97 − 2.97i)21-s + (−3.15 + 3.15i)23-s + (4.42 + 2.32i)25-s − 5.65·27-s + (−3.33 − 3.33i)29-s + 8.92i·31-s + ⋯
L(s)  = 1  + 0.804·3-s + (−0.970 − 0.239i)5-s + (−0.806 − 0.806i)7-s − 0.353·9-s + (−0.654 + 0.654i)11-s + 0.428i·13-s + (−0.780 − 0.192i)15-s + (−0.937 − 0.937i)17-s + (−0.00313 + 0.00313i)19-s + (−0.648 − 0.648i)21-s + (−0.657 + 0.657i)23-s + (0.885 + 0.464i)25-s − 1.08·27-s + (−0.619 − 0.619i)29-s + 1.60i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.990 + 0.137i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.990 + 0.137i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(640\)    =    \(2^{7} \cdot 5\)
Sign: $-0.990 + 0.137i$
Analytic conductor: \(5.11042\)
Root analytic conductor: \(2.26062\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{640} (287, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 640,\ (\ :1/2),\ -0.990 + 0.137i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0172584 - 0.250364i\)
\(L(\frac12)\) \(\approx\) \(0.0172584 - 0.250364i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.17 + 0.535i)T \)
good3 \( 1 - 1.39T + 3T^{2} \)
7 \( 1 + (2.13 + 2.13i)T + 7iT^{2} \)
11 \( 1 + (2.17 - 2.17i)T - 11iT^{2} \)
13 \( 1 - 1.54iT - 13T^{2} \)
17 \( 1 + (3.86 + 3.86i)T + 17iT^{2} \)
19 \( 1 + (0.0136 - 0.0136i)T - 19iT^{2} \)
23 \( 1 + (3.15 - 3.15i)T - 23iT^{2} \)
29 \( 1 + (3.33 + 3.33i)T + 29iT^{2} \)
31 \( 1 - 8.92iT - 31T^{2} \)
37 \( 1 + 7.24iT - 37T^{2} \)
41 \( 1 + 10.3iT - 41T^{2} \)
43 \( 1 + 2.02iT - 43T^{2} \)
47 \( 1 + (-3.34 + 3.34i)T - 47iT^{2} \)
53 \( 1 - 7.30T + 53T^{2} \)
59 \( 1 + (-3.52 - 3.52i)T + 59iT^{2} \)
61 \( 1 + (1.41 - 1.41i)T - 61iT^{2} \)
67 \( 1 + 0.748iT - 67T^{2} \)
71 \( 1 + 0.269T + 71T^{2} \)
73 \( 1 + (-0.811 - 0.811i)T + 73iT^{2} \)
79 \( 1 + 2.80T + 79T^{2} \)
83 \( 1 + 12.8T + 83T^{2} \)
89 \( 1 + 13.3T + 89T^{2} \)
97 \( 1 + (-6.33 - 6.33i)T + 97iT^{2} \)
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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

−10.07444473856430963093577023287, −9.143515289472638495285969575286, −8.507094449670282706311616087586, −7.31355438699805375583304160098, −7.12257944606103505688854398793, −5.48590704669803408939327647864, −4.23986600506811189635023098553, −3.51839698509370798793745985475, −2.33532756461185783855997400737, −0.11313361502350011823030179366, 2.50628399963496160173664956976, 3.18708020432134633776488530506, 4.22538098633987307452226946198, 5.70960535731852953584013770592, 6.49236314949365398895014305914, 7.83129443104884092601535303661, 8.329052008610220799311189828787, 9.026643173792531895202674187814, 10.06566460876696402145729814906, 11.06458393292466645317904025248

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