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Properties

Label 2-640-40.19-c0-0-5
Degree $2$
Conductor $640$
Sign $-0.707 + 0.707i$
Analytic cond. $0.319401$
Root an. cond. $0.565156$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.41i·3-s − i·5-s − 1.41·7-s − 1.00·9-s − 1.41·15-s + 2.00i·21-s + 1.41·23-s − 25-s + 1.41i·35-s − 1.41i·43-s + 1.00i·45-s + 1.41·47-s + 1.00·49-s − 2i·61-s + 1.41·63-s + ⋯
L(s)  = 1  − 1.41i·3-s − i·5-s − 1.41·7-s − 1.00·9-s − 1.41·15-s + 2.00i·21-s + 1.41·23-s − 25-s + 1.41i·35-s − 1.41i·43-s + 1.00i·45-s + 1.41·47-s + 1.00·49-s − 2i·61-s + 1.41·63-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(640\)    =    \(2^{7} \cdot 5\)
Sign: $-0.707 + 0.707i$
Analytic conductor: \(0.319401\)
Root analytic conductor: \(0.565156\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{640} (319, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 640,\ (\ :0),\ -0.707 + 0.707i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7614277285\)
\(L(\frac12)\) \(\approx\) \(0.7614277285\)
\(L(1)\) 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 \( 1 + iT \)
good3 \( 1 + 1.41iT - T^{2} \)
7 \( 1 + 1.41T + T^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 - 1.41T + T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + 1.41iT - T^{2} \)
47 \( 1 - 1.41T + T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 + 2iT - T^{2} \)
67 \( 1 - 1.41iT - T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - 1.41iT - T^{2} \)
89 \( 1 - 2T + T^{2} \)
97 \( 1 - T^{2} \)
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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

−10.40967131548570777254512737267, −9.353676599542438581069966215702, −8.746197053727926719614644644105, −7.69239007266156198690974115550, −6.90201687904628848868726178044, −6.16129301473884797456022395568, −5.15384715775471805801664058613, −3.67698404488234637029713469462, −2.36167708283086697858559915753, −0.888968719348212404630780880676, 2.84036874479841113515575374120, 3.44855563906438548546424478267, 4.47045867479028155561595847397, 5.70190271462894418090065379830, 6.57696853879096002680278958685, 7.45761728779374427310111341324, 8.948348040128907045377525178163, 9.511032070207247755409928120889, 10.28230857549234278855753994928, 10.75969005736860825951260313667

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