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Properties

Label 2-640-80.3-c1-0-1
Degree $2$
Conductor $640$
Sign $-0.977 - 0.209i$
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  − 0.692·3-s + (0.245 + 2.22i)5-s + (−0.343 + 0.343i)7-s − 2.52·9-s + (−0.843 − 0.843i)11-s + 3.68i·13-s + (−0.169 − 1.53i)15-s + (0.412 − 0.412i)17-s + (−5.37 − 5.37i)19-s + (0.238 − 0.238i)21-s + (−3.08 − 3.08i)23-s + (−4.87 + 1.09i)25-s + 3.82·27-s + (−4.22 + 4.22i)29-s + 8.75i·31-s + ⋯
L(s)  = 1  − 0.399·3-s + (0.109 + 0.993i)5-s + (−0.129 + 0.129i)7-s − 0.840·9-s + (−0.254 − 0.254i)11-s + 1.02i·13-s + (−0.0438 − 0.397i)15-s + (0.0999 − 0.0999i)17-s + (−1.23 − 1.23i)19-s + (0.0519 − 0.0519i)21-s + (−0.643 − 0.643i)23-s + (−0.975 + 0.218i)25-s + 0.735·27-s + (−0.785 + 0.785i)29-s + 1.57i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.977 - 0.209i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.977 - 0.209i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(640\)    =    \(2^{7} \cdot 5\)
Sign: $-0.977 - 0.209i$
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: \((2,\ 640,\ (\ :1/2),\ -0.977 - 0.209i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0428872 + 0.405745i\)
\(L(\frac12)\) \(\approx\) \(0.0428872 + 0.405745i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.245 - 2.22i)T \)
good3 \( 1 + 0.692T + 3T^{2} \)
7 \( 1 + (0.343 - 0.343i)T - 7iT^{2} \)
11 \( 1 + (0.843 + 0.843i)T + 11iT^{2} \)
13 \( 1 - 3.68iT - 13T^{2} \)
17 \( 1 + (-0.412 + 0.412i)T - 17iT^{2} \)
19 \( 1 + (5.37 + 5.37i)T + 19iT^{2} \)
23 \( 1 + (3.08 + 3.08i)T + 23iT^{2} \)
29 \( 1 + (4.22 - 4.22i)T - 29iT^{2} \)
31 \( 1 - 8.75iT - 31T^{2} \)
37 \( 1 + 5.41iT - 37T^{2} \)
41 \( 1 - 2.54iT - 41T^{2} \)
43 \( 1 + 4.30iT - 43T^{2} \)
47 \( 1 + (4.56 + 4.56i)T + 47iT^{2} \)
53 \( 1 + 6.07T + 53T^{2} \)
59 \( 1 + (7.33 - 7.33i)T - 59iT^{2} \)
61 \( 1 + (-4.81 - 4.81i)T + 61iT^{2} \)
67 \( 1 - 14.3iT - 67T^{2} \)
71 \( 1 + 2.97T + 71T^{2} \)
73 \( 1 + (-6.87 + 6.87i)T - 73iT^{2} \)
79 \( 1 + 10.1T + 79T^{2} \)
83 \( 1 - 7.15T + 83T^{2} \)
89 \( 1 + 1.10T + 89T^{2} \)
97 \( 1 + (-7.15 + 7.15i)T - 97iT^{2} \)
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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

−10.94263642723924699806537775366, −10.40205855401260547254308008994, −9.159325192877295710546796163361, −8.520049305669413695494838242157, −7.19221883292743047737600505351, −6.53045161830102510532533989356, −5.72145400954093233254535394646, −4.54362404855978736355570672260, −3.22316547326679037949274248092, −2.20561981532157022062344567129, 0.21523756512970331191246582585, 1.97753913166018453046786636364, 3.55872193707723186091041134896, 4.70210014446383108891232821175, 5.71592002354942180135551586489, 6.20375118502069654844580567909, 8.001654032866893625091131922616, 8.093613754868309211621642097835, 9.426809252534299737834402390840, 10.09218805007617161378346280999

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