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Properties

Label 1-640-640.187-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  + (0.555 − 0.831i)3-s + (0.923 + 0.382i)7-s + (−0.382 − 0.923i)9-s + (−0.555 − 0.831i)11-s + (−0.195 − 0.980i)13-s + (0.707 + 0.707i)17-s + (0.195 + 0.980i)19-s + (0.831 − 0.555i)21-s + (−0.382 − 0.923i)23-s + (−0.980 − 0.195i)27-s + (0.555 − 0.831i)29-s − i·31-s − 33-s + (−0.980 − 0.195i)37-s + (−0.923 − 0.382i)39-s + ⋯
L(s)  = 1  + (0.555 − 0.831i)3-s + (0.923 + 0.382i)7-s + (−0.382 − 0.923i)9-s + (−0.555 − 0.831i)11-s + (−0.195 − 0.980i)13-s + (0.707 + 0.707i)17-s + (0.195 + 0.980i)19-s + (0.831 − 0.555i)21-s + (−0.382 − 0.923i)23-s + (−0.980 − 0.195i)27-s + (0.555 − 0.831i)29-s − i·31-s − 33-s + (−0.980 − 0.195i)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: $\chi_{640} (187, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 640,\ (0:\ ),\ 0.0136 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.238313739 - 1.221570131i\)
\(L(\frac12)\) \(\approx\) \(1.238313739 - 1.221570131i\)
\(L(1)\) \(\approx\) \(1.220739478 - 0.5182415307i\)
\(L(1)\) \(\approx\) \(1.220739478 - 0.5182415307i\)

Euler product

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

−23.22431916753504336946878169922, −22.03126329889874144259568391219, −21.370809109975608791923826046604, −20.71864160821592185611155121628, −20.00594557108719578776604876805, −19.202140576169568952485294061084, −18.01901628878706827559126052345, −17.37923128118155378644828172334, −16.27440289356125235974576481310, −15.695955489053998945575005471318, −14.63938948623417209537485622663, −14.16699706850828269102469135071, −13.3119785234936332813411340258, −11.98463456005145347368256789750, −11.20984286882469426536853647731, −10.27867863519941538916193968141, −9.535600152037778102254692937919, −8.648095437839982545880868667423, −7.671243316379763737413933662782, −6.956343855318058871265144859003, −5.140799096270448118301797071208, −4.8462816248817088762344233509, −3.73888595617254958148584144206, −2.6177197467169889042464617751, −1.56469091222120936281672538420, 0.83840261009821719814935236802, 2.036496053035344318161834371406, 2.91963353192088124571084981411, 4.02376878257563000487307221185, 5.53317023187544418294266955431, 6.023430814647700274040365097795, 7.48554308046294744101043902366, 8.13464473204906415630521815874, 8.59447983609936436919540814388, 9.97723544152017323672394467895, 10.87208619973170439868172324813, 11.99924562962869604743818274270, 12.554445520504506412576255399181, 13.56046301539026451733282728816, 14.3503602158802125819039259158, 14.97832928826983636241958189580, 15.961565265805326085281375836179, 17.194340452905873917915526665215, 17.82637760056905496700744877005, 18.81724929346644565208557643903, 19.07282936329262517859384718311, 20.51374502623318716142397913927, 20.733614439548969691690332059983, 21.7947990115335676134320350545, 22.856162129325108447141047953894

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