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Properties

Label 1-640-640.499-r1-0-0
Degree $1$
Conductor $640$
Sign $0.857 + 0.514i$
Analytic cond. $68.7775$
Root an. cond. $68.7775$
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  + (−0.831 + 0.555i)3-s + (−0.382 − 0.923i)7-s + (0.382 − 0.923i)9-s + (−0.555 + 0.831i)11-s + (−0.980 − 0.195i)13-s + (−0.707 − 0.707i)17-s + (−0.195 + 0.980i)19-s + (0.831 + 0.555i)21-s + (−0.923 − 0.382i)23-s + (0.195 + 0.980i)27-s + (−0.555 − 0.831i)29-s + i·31-s − i·33-s + (0.195 + 0.980i)37-s + (0.923 − 0.382i)39-s + ⋯
L(s)  = 1  + (−0.831 + 0.555i)3-s + (−0.382 − 0.923i)7-s + (0.382 − 0.923i)9-s + (−0.555 + 0.831i)11-s + (−0.980 − 0.195i)13-s + (−0.707 − 0.707i)17-s + (−0.195 + 0.980i)19-s + (0.831 + 0.555i)21-s + (−0.923 − 0.382i)23-s + (0.195 + 0.980i)27-s + (−0.555 − 0.831i)29-s + i·31-s − i·33-s + (0.195 + 0.980i)37-s + (0.923 − 0.382i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.857 + 0.514i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.857 + 0.514i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(640\)    =    \(2^{7} \cdot 5\)
Sign: $0.857 + 0.514i$
Analytic conductor: \(68.7775\)
Root analytic conductor: \(68.7775\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{640} (499, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 640,\ (1:\ ),\ 0.857 + 0.514i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6170485799 + 0.1707603395i\)
\(L(\frac12)\) \(\approx\) \(0.6170485799 + 0.1707603395i\)
\(L(1)\) \(\approx\) \(0.6150690228 + 0.04758105052i\)
\(L(1)\) \(\approx\) \(0.6150690228 + 0.04758105052i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
good3 \( 1 + (-0.831 + 0.555i)T \)
7 \( 1 + (-0.382 - 0.923i)T \)
11 \( 1 + (-0.555 + 0.831i)T \)
13 \( 1 + (-0.980 - 0.195i)T \)
17 \( 1 + (-0.707 - 0.707i)T \)
19 \( 1 + (-0.195 + 0.980i)T \)
23 \( 1 + (-0.923 - 0.382i)T \)
29 \( 1 + (-0.555 - 0.831i)T \)
31 \( 1 + iT \)
37 \( 1 + (0.195 + 0.980i)T \)
41 \( 1 + (0.923 + 0.382i)T \)
43 \( 1 + (-0.831 - 0.555i)T \)
47 \( 1 + (-0.707 - 0.707i)T \)
53 \( 1 + (0.555 - 0.831i)T \)
59 \( 1 + (-0.980 + 0.195i)T \)
61 \( 1 + (0.831 - 0.555i)T \)
67 \( 1 + (0.831 - 0.555i)T \)
71 \( 1 + (-0.382 - 0.923i)T \)
73 \( 1 + (0.382 - 0.923i)T \)
79 \( 1 + (0.707 - 0.707i)T \)
83 \( 1 + (-0.195 + 0.980i)T \)
89 \( 1 + (-0.923 + 0.382i)T \)
97 \( 1 - iT \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−22.479856252031629728347282146467, −21.86175109464359431447287326660, −21.44307672045372864897313952021, −19.84235067796739888776801050802, −19.28145241388774856770596966165, −18.444592754264854303945229520862, −17.769510761579876892672275977902, −16.895876913015070282262571622209, −16.04684519468573058716112712849, −15.34373046842503093011860279327, −14.20647666433881726535545620561, −13.03916846713808755420180740775, −12.69671362019210613721330807370, −11.585980228657626387952070025900, −11.03998632084868029680075740806, −9.929330057965175998509454914892, −8.91813256695374089798895759490, −7.92205669375465071530945532149, −6.95221583753901301597815342073, −5.99412221984061778326283247703, −5.41102745330379200476574220348, −4.29129278780568896884799994449, −2.74638874287230853693354212715, −1.944703811291467900029711369192, −0.35688167647968666630794280086, 0.46490366332915821677568298051, 2.038449352627934766778580918, 3.452838071376145487730359458030, 4.45971756673194124493838564024, 5.09097545964741009682438493216, 6.32057926612630442282951193980, 7.08625532533668348372241217572, 8.02553899716900106910494908067, 9.58024974443955028282089952753, 10.0278646161439858082009702141, 10.74952927855056451372328472977, 11.84755034648712449875601182398, 12.539364612277415317683816808945, 13.47703353885371381111667035862, 14.58861403883565722532172529412, 15.4008916797451255222596909544, 16.287087901425596419706935214581, 16.90394167018682148674279961886, 17.730748690530817962360175950212, 18.40008815375174812475953759043, 19.70672732514952496882394836310, 20.38115934258198478687659818027, 21.10715536509666381024858759154, 22.19487065607963483272600481833, 22.73816759923418805988977343655

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