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Properties

Label 2-640-8.5-c1-0-8
Degree $2$
Conductor $640$
Sign $0.707 + 0.707i$
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  − 1.41i·3-s − i·5-s + 4.24·7-s + 0.999·9-s + 5.65i·11-s − 2i·13-s − 1.41·15-s + 6·17-s + 2.82i·19-s − 6i·21-s − 7.07·23-s − 25-s − 5.65i·27-s − 4i·29-s + 2.82·31-s + ⋯
L(s)  = 1  − 0.816i·3-s − 0.447i·5-s + 1.60·7-s + 0.333·9-s + 1.70i·11-s − 0.554i·13-s − 0.365·15-s + 1.45·17-s + 0.648i·19-s − 1.30i·21-s − 1.47·23-s − 0.200·25-s − 1.08i·27-s − 0.742i·29-s + 0.508·31-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(640\)    =    \(2^{7} \cdot 5\)
Sign: $0.707 + 0.707i$
Analytic conductor: \(5.11042\)
Root analytic conductor: \(2.26062\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{640} (321, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 640,\ (\ :1/2),\ 0.707 + 0.707i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.71723 - 0.711300i\)
\(L(\frac12)\) \(\approx\) \(1.71723 - 0.711300i\)
\(L(\frac{3}{2})\) 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 \( 1 + iT \)
good3 \( 1 + 1.41iT - 3T^{2} \)
7 \( 1 - 4.24T + 7T^{2} \)
11 \( 1 - 5.65iT - 11T^{2} \)
13 \( 1 + 2iT - 13T^{2} \)
17 \( 1 - 6T + 17T^{2} \)
19 \( 1 - 2.82iT - 19T^{2} \)
23 \( 1 + 7.07T + 23T^{2} \)
29 \( 1 + 4iT - 29T^{2} \)
31 \( 1 - 2.82T + 31T^{2} \)
37 \( 1 + 2iT - 37T^{2} \)
41 \( 1 + 8T + 41T^{2} \)
43 \( 1 + 1.41iT - 43T^{2} \)
47 \( 1 - 1.41T + 47T^{2} \)
53 \( 1 - 2iT - 53T^{2} \)
59 \( 1 - 2.82iT - 59T^{2} \)
61 \( 1 + 14iT - 61T^{2} \)
67 \( 1 - 4.24iT - 67T^{2} \)
71 \( 1 + 2.82T + 71T^{2} \)
73 \( 1 + 6T + 73T^{2} \)
79 \( 1 + 16.9T + 79T^{2} \)
83 \( 1 - 12.7iT - 83T^{2} \)
89 \( 1 + 6T + 89T^{2} \)
97 \( 1 - 10T + 97T^{2} \)
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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

−10.22485603313976737998706892983, −9.860226286727572560246412756713, −8.307578026007339622648312811243, −7.83009231503212342238693861276, −7.24182939589077284796216240559, −5.88570515159656489678736126700, −4.92164590112434797275952424343, −4.08198216291624469518957446726, −2.08244044112979950983335092636, −1.36526500724311728966008174151, 1.47020581996749346135711012764, 3.15043352519687779289810720227, 4.15515308599506132677908864972, 5.11113362790679925866945707521, 5.97626124721327393839958169505, 7.29314904572308201147759402196, 8.190786937056095097840630135205, 8.853864085037914397617140435488, 10.06181149565975523129948369720, 10.60358626758620028951560726615

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