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Properties

Label 1-640-640.427-r0-0-0
Degree $1$
Conductor $640$
Sign $0.395 - 0.918i$
Analytic cond. $2.97214$
Root an. cond. $2.97214$
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  + (0.980 − 0.195i)3-s + (0.382 − 0.923i)7-s + (0.923 − 0.382i)9-s + (−0.980 − 0.195i)11-s + (−0.555 + 0.831i)13-s + (−0.707 − 0.707i)17-s + (0.555 − 0.831i)19-s + (0.195 − 0.980i)21-s + (0.923 − 0.382i)23-s + (0.831 − 0.555i)27-s + (0.980 − 0.195i)29-s − i·31-s − 33-s + (0.831 − 0.555i)37-s + (−0.382 + 0.923i)39-s + ⋯
L(s)  = 1  + (0.980 − 0.195i)3-s + (0.382 − 0.923i)7-s + (0.923 − 0.382i)9-s + (−0.980 − 0.195i)11-s + (−0.555 + 0.831i)13-s + (−0.707 − 0.707i)17-s + (0.555 − 0.831i)19-s + (0.195 − 0.980i)21-s + (0.923 − 0.382i)23-s + (0.831 − 0.555i)27-s + (0.980 − 0.195i)29-s − i·31-s − 33-s + (0.831 − 0.555i)37-s + (−0.382 + 0.923i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.395 - 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.395 - 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(640\)    =    \(2^{7} \cdot 5\)
Sign: $0.395 - 0.918i$
Analytic conductor: \(2.97214\)
Root analytic conductor: \(2.97214\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{640} (427, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 640,\ (0:\ ),\ 0.395 - 0.918i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.601308605 - 1.054265757i\)
\(L(\frac12)\) \(\approx\) \(1.601308605 - 1.054265757i\)
\(L(1)\) \(\approx\) \(1.389020886 - 0.3834304798i\)
\(L(1)\) \(\approx\) \(1.389020886 - 0.3834304798i\)

Euler product

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

−23.05738988954123513498602902619, −21.96061689643273234412514136254, −21.38765440113051337274213789220, −20.61727458735294503759647501577, −19.83539016869550925163037919265, −19.00577831686129462394476824378, −18.20815303459127033306554419086, −17.49151942088677742705253734874, −16.09737696028628124812495884910, −15.42638484058442432298166329289, −14.86727753202446129982615358789, −13.99441002028167225360491365931, −12.903103357988170163817016783924, −12.44508826860526305025286111585, −11.09312772682791909505576683063, −10.18983078012152132567926854743, −9.4037244920971656327032206442, −8.29227237066889370237658044480, −7.97731953294251843572797384892, −6.75264522088068029076164550024, −5.37537868575798616334229227946, −4.74289575384168585779128826007, −3.31822610422178969928167056417, −2.60950008989111683034676320817, −1.59815494868060993466974889674, 0.865730383765952652965856078362, 2.25861622412022184845136975958, 3.00145053846501940538633559224, 4.32281581822653313811587656365, 4.91446275956542248104747056788, 6.62866310748349563205458833755, 7.32595968638344088031732976510, 8.06146292537070404120901644387, 9.08767497988097260390740147406, 9.85129494085208877293008229123, 10.863580466467417075272585750044, 11.74869161749552507453064402330, 13.09214691827154589246249112094, 13.505744691654372469859040093009, 14.31606537541116879390482974193, 15.16005632364390930119377973438, 16.02910437720022675976358122076, 16.94334606247634274341445467617, 18.02247882757726176774570771601, 18.62177860179601495836371491392, 19.72683617089282653844585171031, 20.13352778809758167082302533891, 21.092831666837449008806125954986, 21.61524757397413004742520262198, 22.91355125395114726449889718998

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