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Properties

Label 1-671-671.10-r0-0-0
Degree $1$
Conductor $671$
Sign $-0.352 - 0.935i$
Analytic cond. $3.11611$
Root an. cond. $3.11611$
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  + (0.743 − 0.669i)2-s + (−0.309 + 0.951i)3-s + (0.104 − 0.994i)4-s + (−0.913 + 0.406i)5-s + (0.406 + 0.913i)6-s + (−0.207 + 0.978i)7-s + (−0.587 − 0.809i)8-s + (−0.809 − 0.587i)9-s + (−0.406 + 0.913i)10-s + (0.913 + 0.406i)12-s + (0.5 − 0.866i)13-s + (0.5 + 0.866i)14-s + (−0.104 − 0.994i)15-s + (−0.978 − 0.207i)16-s + (−0.994 − 0.104i)17-s + (−0.994 + 0.104i)18-s + ⋯
L(s)  = 1  + (0.743 − 0.669i)2-s + (−0.309 + 0.951i)3-s + (0.104 − 0.994i)4-s + (−0.913 + 0.406i)5-s + (0.406 + 0.913i)6-s + (−0.207 + 0.978i)7-s + (−0.587 − 0.809i)8-s + (−0.809 − 0.587i)9-s + (−0.406 + 0.913i)10-s + (0.913 + 0.406i)12-s + (0.5 − 0.866i)13-s + (0.5 + 0.866i)14-s + (−0.104 − 0.994i)15-s + (−0.978 − 0.207i)16-s + (−0.994 − 0.104i)17-s + (−0.994 + 0.104i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 671 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.352 - 0.935i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 671 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.352 - 0.935i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(671\)    =    \(11 \cdot 61\)
Sign: $-0.352 - 0.935i$
Analytic conductor: \(3.11611\)
Root analytic conductor: \(3.11611\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{671} (10, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 671,\ (0:\ ),\ -0.352 - 0.935i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4758423798 - 0.6877334547i\)
\(L(\frac12)\) \(\approx\) \(0.4758423798 - 0.6877334547i\)
\(L(1)\) \(\approx\) \(0.9427137785 - 0.2031550437i\)
\(L(1)\) \(\approx\) \(0.9427137785 - 0.2031550437i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
61 \( 1 \)
good2 \( 1 + (0.743 - 0.669i)T \)
3 \( 1 + (-0.309 + 0.951i)T \)
5 \( 1 + (-0.913 + 0.406i)T \)
7 \( 1 + (-0.207 + 0.978i)T \)
13 \( 1 + (0.5 - 0.866i)T \)
17 \( 1 + (-0.994 - 0.104i)T \)
19 \( 1 + (-0.978 + 0.207i)T \)
23 \( 1 + (0.587 - 0.809i)T \)
29 \( 1 + (0.866 - 0.5i)T \)
31 \( 1 + (-0.743 - 0.669i)T \)
37 \( 1 + (0.951 - 0.309i)T \)
41 \( 1 + (0.309 + 0.951i)T \)
43 \( 1 + (0.994 - 0.104i)T \)
47 \( 1 + (-0.5 - 0.866i)T \)
53 \( 1 + (-0.587 - 0.809i)T \)
59 \( 1 + (0.743 - 0.669i)T \)
67 \( 1 + (-0.406 - 0.913i)T \)
71 \( 1 + (0.406 - 0.913i)T \)
73 \( 1 + (-0.913 - 0.406i)T \)
79 \( 1 + (-0.994 + 0.104i)T \)
83 \( 1 + (-0.669 - 0.743i)T \)
89 \( 1 + (-0.951 - 0.309i)T \)
97 \( 1 + (-0.669 + 0.743i)T \)
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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

−23.37709877228655918905701283108, −22.59203277295093134840667761842, −21.57732400583149959724229177434, −20.529269232750012988054438366772, −19.722328230854313561044267307460, −19.0720443577434201644297766508, −17.80348511640086660187323112361, −17.156413320199467763106261533220, −16.37230582056898187861506073863, −15.74662427632108086904193538525, −14.58111381244894333114431012035, −13.783355338886582170268986512658, −12.9990243202658995789516745041, −12.49978238427744717551751607324, −11.34261140202374555984360260821, −10.98330917320266090988580038361, −8.96380634857712436968165600424, −8.30235755056859837804475647771, −7.201572972885028225535423858144, −6.92055773258525896031498352647, −5.87851960815671552179885580979, −4.61256713689103118587663784077, −4.036779513996432249299874233196, −2.81177233997658354174709103245, −1.31346578328976977798522654883, 0.342532386307333377886535126656, 2.40972886564573140435851931641, 3.14467967315376641285583885118, 4.11317662580534827080942272626, 4.8152474484755807674955714713, 5.93061011751244623723016586244, 6.55874015814777551788813495789, 8.26422831273186596478930236609, 9.08271769558077555353274229405, 10.140015647995423862134833687455, 10.99285010677088725082506928091, 11.41085191098560702945159608114, 12.40818089757177861533759273585, 13.08660102156345442225813490558, 14.57529739450276638610814974962, 15.02621853698914609780667498666, 15.657397726230280004925048662439, 16.32997233446465575260621820394, 17.8035101741402006177877815668, 18.59160783795692817123040054055, 19.49415814388806774398960609178, 20.19011913332518611375058125547, 21.03988174255045703837378882048, 21.779639348830469673648133267766, 22.55305564324766718951341407942

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