Properties

Label 1-643-643.12-r1-0-0
Degree $1$
Conductor $643$
Sign $-0.405 + 0.914i$
Analytic cond. $69.0999$
Root an. cond. $69.0999$
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  + (−0.386 + 0.922i)2-s + (0.246 − 0.969i)3-s + (−0.701 − 0.712i)4-s + (0.815 + 0.578i)5-s + (0.798 + 0.601i)6-s + (0.972 + 0.232i)7-s + (0.928 − 0.372i)8-s + (−0.877 − 0.478i)9-s + (−0.848 + 0.529i)10-s + (0.780 − 0.625i)11-s + (−0.863 + 0.504i)12-s + (−0.984 − 0.175i)13-s + (−0.590 + 0.807i)14-s + (0.761 − 0.647i)15-s + (−0.0146 + 0.999i)16-s + (−0.832 + 0.554i)17-s + ⋯
L(s)  = 1  + (−0.386 + 0.922i)2-s + (0.246 − 0.969i)3-s + (−0.701 − 0.712i)4-s + (0.815 + 0.578i)5-s + (0.798 + 0.601i)6-s + (0.972 + 0.232i)7-s + (0.928 − 0.372i)8-s + (−0.877 − 0.478i)9-s + (−0.848 + 0.529i)10-s + (0.780 − 0.625i)11-s + (−0.863 + 0.504i)12-s + (−0.984 − 0.175i)13-s + (−0.590 + 0.807i)14-s + (0.761 − 0.647i)15-s + (−0.0146 + 0.999i)16-s + (−0.832 + 0.554i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 643 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.405 + 0.914i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 643 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.405 + 0.914i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(643\)
Sign: $-0.405 + 0.914i$
Analytic conductor: \(69.0999\)
Root analytic conductor: \(69.0999\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{643} (12, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 643,\ (1:\ ),\ -0.405 + 0.914i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8710627413 + 1.339203620i\)
\(L(\frac12)\) \(\approx\) \(0.8710627413 + 1.339203620i\)
\(L(1)\) \(\approx\) \(0.9892707632 + 0.3299366331i\)
\(L(1)\) \(\approx\) \(0.9892707632 + 0.3299366331i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad643 \( 1 \)
good2 \( 1 + (-0.386 + 0.922i)T \)
3 \( 1 + (0.246 - 0.969i)T \)
5 \( 1 + (0.815 + 0.578i)T \)
7 \( 1 + (0.972 + 0.232i)T \)
11 \( 1 + (0.780 - 0.625i)T \)
13 \( 1 + (-0.984 - 0.175i)T \)
17 \( 1 + (-0.832 + 0.554i)T \)
19 \( 1 + (0.904 + 0.426i)T \)
23 \( 1 + (-0.516 + 0.856i)T \)
29 \( 1 + (-0.780 + 0.625i)T \)
31 \( 1 + (0.761 + 0.647i)T \)
37 \( 1 + (-0.938 + 0.345i)T \)
41 \( 1 + (-0.160 + 0.986i)T \)
43 \( 1 + (-0.891 + 0.452i)T \)
47 \( 1 + (-0.972 - 0.232i)T \)
53 \( 1 + (0.722 + 0.691i)T \)
59 \( 1 + (0.246 + 0.969i)T \)
61 \( 1 + (0.131 - 0.991i)T \)
67 \( 1 + (0.965 - 0.261i)T \)
71 \( 1 + (-0.957 - 0.289i)T \)
73 \( 1 + (0.131 - 0.991i)T \)
79 \( 1 + (-0.957 + 0.289i)T \)
83 \( 1 + (0.761 - 0.647i)T \)
89 \( 1 + (0.386 + 0.922i)T \)
97 \( 1 + (0.491 + 0.870i)T \)
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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

−22.22664842611135171353723060968, −21.45935750866437226351310612132, −20.51052433926452134860827479152, −20.40006664830017641579910285598, −19.48473019946396887063793098127, −18.17245189551686340856999363821, −17.28841880571613153545339857663, −17.10236145533383523599242719871, −15.956091519754194797374599541972, −14.67281408265884209943134225916, −14.05869944864939538342625050114, −13.26223524352580782493898120972, −11.960639639518244712520944498913, −11.479089480334918964893530416238, −10.30275654334090769215981999315, −9.73028903608245316745843832415, −9.031089883860547603133346178598, −8.27031205843030401108459008434, −7.076471214471838122162366943301, −5.29966263970740246416427444685, −4.68461075061707322924880091396, −3.98110493305054482644230701262, −2.47475477560957841248122198598, −1.86261863298864084225508253151, −0.40925884030452060613197334843, 1.28038432474239934353223341402, 1.914332957490720411576110558941, 3.34705929174865662971198685184, 4.9846912568855645346634805745, 5.81000003481093903644547769414, 6.609071041420599432626235480039, 7.393728087496923730312749656291, 8.24940123208809180168429275606, 9.04888772121567763601198620674, 9.95953277801346283355733403633, 11.11450082598115030088969302664, 12.012827736112793860794309262652, 13.328685992203845113211852532077, 13.92710964996482598080156426456, 14.59702485730353488153621699015, 15.15351598347894697507876707333, 16.625442766807949477100504802613, 17.45643116464532753330262562367, 17.84499351358774848322463095750, 18.570835299671131404976548951134, 19.4278167546783236501426934874, 20.16131019300612899016128744533, 21.6082999096033089235442150462, 22.20279240269339859820781623790, 23.15318957913442014965329176509

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