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Properties

Label 1-640-640.347-r0-0-0
Degree $1$
Conductor $640$
Sign $0.0136 - 0.999i$
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  + (−0.831 − 0.555i)3-s + (−0.923 − 0.382i)7-s + (0.382 + 0.923i)9-s + (0.831 − 0.555i)11-s + (−0.980 + 0.195i)13-s + (0.707 + 0.707i)17-s + (0.980 − 0.195i)19-s + (0.555 + 0.831i)21-s + (0.382 + 0.923i)23-s + (0.195 − 0.980i)27-s + (−0.831 − 0.555i)29-s − i·31-s − 33-s + (0.195 − 0.980i)37-s + (0.923 + 0.382i)39-s + ⋯
L(s)  = 1  + (−0.831 − 0.555i)3-s + (−0.923 − 0.382i)7-s + (0.382 + 0.923i)9-s + (0.831 − 0.555i)11-s + (−0.980 + 0.195i)13-s + (0.707 + 0.707i)17-s + (0.980 − 0.195i)19-s + (0.555 + 0.831i)21-s + (0.382 + 0.923i)23-s + (0.195 − 0.980i)27-s + (−0.831 − 0.555i)29-s − i·31-s − 33-s + (0.195 − 0.980i)37-s + (0.923 + 0.382i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0136 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0136 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(640\)    =    \(2^{7} \cdot 5\)
Sign: $0.0136 - 0.999i$
Analytic conductor: \(2.97214\)
Root analytic conductor: \(2.97214\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{640} (347, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 640,\ (0:\ ),\ 0.0136 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5779473684 - 0.5701327705i\)
\(L(\frac12)\) \(\approx\) \(0.5779473684 - 0.5701327705i\)
\(L(1)\) \(\approx\) \(0.7261373436 - 0.2199777026i\)
\(L(1)\) \(\approx\) \(0.7261373436 - 0.2199777026i\)

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 \( 1 + (-0.923 - 0.382i)T \)
11 \( 1 + (0.831 - 0.555i)T \)
13 \( 1 + (-0.980 + 0.195i)T \)
17 \( 1 + (0.707 + 0.707i)T \)
19 \( 1 + (0.980 - 0.195i)T \)
23 \( 1 + (0.382 + 0.923i)T \)
29 \( 1 + (-0.831 - 0.555i)T \)
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 \( 1 + (-0.195 + 0.980i)T \)
61 \( 1 + (0.555 - 0.831i)T \)
67 \( 1 + (-0.831 - 0.555i)T \)
71 \( 1 + (0.382 - 0.923i)T \)
73 \( 1 + (0.923 - 0.382i)T \)
79 \( 1 + (-0.707 - 0.707i)T \)
83 \( 1 + (-0.195 - 0.980i)T \)
89 \( 1 + (-0.923 - 0.382i)T \)
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

−22.73608775555242222980761775644, −22.42260319741456945180507897352, −21.7821695920375648026347264365, −20.616279496923895063839105417036, −19.980056565109161536520592759740, −18.863670281107512104902340906888, −18.159782334404320058551746016940, −17.088853326079599092395255792058, −16.59131160068347378980716065221, −15.75046286665478684857231241879, −14.92875270556736022684506329334, −14.08449453927027425145264321200, −12.62103050029412875935252727964, −12.24032497757342705659794448746, −11.40817186904473972628893370574, −10.16258675352765374777851321540, −9.69800273269286332466686696711, −8.9018355893563326774269135313, −7.28050484434976359942313969464, −6.678307583408372595923542246934, −5.56818697809615793520239967484, −4.87223763117714297986780872774, −3.70823036874724734883885569790, −2.76426324774212504529598650729, −1.0770742051670051281797516602, 0.549035674036149578922537626723, 1.75855945978361399472334173238, 3.17637274950743235061192012369, 4.198187350924988859000107944, 5.501434079404735448249643438, 6.135789033597618160799240642308, 7.158693699402883979833562472192, 7.72497129909419986344279930348, 9.26784245180522094359246565330, 9.91700926382438882787313144896, 11.00937521492818027782329181064, 11.79726537509590271999227037303, 12.54863672060469679891479892801, 13.38901960652546838112810493771, 14.1574628525507199092681377214, 15.33145297245108674962393350104, 16.39658594891209461876067733875, 16.92397396802346197790203131438, 17.538840740746841284908539425643, 18.78833838476160788385759439490, 19.26838886225462759479028675818, 19.962022446069110478718801442862, 21.27505780473329886811368539327, 22.27075285774776856051563969109, 22.48655854417994244063343745111

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