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Properties

Label 1-671-671.136-r0-0-0
Degree $1$
Conductor $671$
Sign $0.123 + 0.992i$
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  + (0.978 + 0.207i)2-s + (−0.809 − 0.587i)3-s + (0.913 + 0.406i)4-s + (0.669 + 0.743i)5-s + (−0.669 − 0.743i)6-s + (0.104 + 0.994i)7-s + (0.809 + 0.587i)8-s + (0.309 + 0.951i)9-s + (0.5 + 0.866i)10-s + (−0.5 − 0.866i)12-s + (−0.978 − 0.207i)13-s + (−0.104 + 0.994i)14-s + (−0.104 − 0.994i)15-s + (0.669 + 0.743i)16-s + (0.978 − 0.207i)17-s + (0.104 + 0.994i)18-s + ⋯
L(s)  = 1  + (0.978 + 0.207i)2-s + (−0.809 − 0.587i)3-s + (0.913 + 0.406i)4-s + (0.669 + 0.743i)5-s + (−0.669 − 0.743i)6-s + (0.104 + 0.994i)7-s + (0.809 + 0.587i)8-s + (0.309 + 0.951i)9-s + (0.5 + 0.866i)10-s + (−0.5 − 0.866i)12-s + (−0.978 − 0.207i)13-s + (−0.104 + 0.994i)14-s + (−0.104 − 0.994i)15-s + (0.669 + 0.743i)16-s + (0.978 − 0.207i)17-s + (0.104 + 0.994i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 671 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.123 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 671 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.123 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(671\)    =    \(11 \cdot 61\)
Sign: $0.123 + 0.992i$
Analytic conductor: \(3.11611\)
Root analytic conductor: \(3.11611\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{671} (136, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 671,\ (0:\ ),\ 0.123 + 0.992i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.625593863 + 1.436059458i\)
\(L(\frac12)\) \(\approx\) \(1.625593863 + 1.436059458i\)
\(L(1)\) \(\approx\) \(1.533691790 + 0.5357076941i\)
\(L(1)\) \(\approx\) \(1.533691790 + 0.5357076941i\)

Euler product

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

−22.344096384079963787916161691806, −21.86059603651232387493781520522, −21.146964320947355383309780360832, −20.31716577200379146012720753934, −19.83920338485988878628799971103, −18.39904577186291215487106639101, −17.18354757145373154987144304470, −16.803075922159260946566617612269, −16.12130138190153753661694632516, −14.98402616592082115314961339946, −14.27168142934042695609321271784, −13.28970944177937518544172230263, −12.60549300948009372003704828329, −11.743848652105091722710660744657, −10.91865361100102515462175149974, −9.97073480419662557482063215870, −9.53679143974037861800330716570, −7.78930808815343853326711616513, −6.76752511765534160768623884112, −5.86685291482018985329765423044, −5.01730170105209677721716416088, −4.43496174822695324606681044066, −3.47775389644229664195222609419, −1.99655969542547966725355728934, −0.82205937248842165101782225210, 1.82004405757978265367447687163, 2.39231803554145215036132797046, 3.60935982801972758944137751505, 5.12562665433090425033828650728, 5.64742746790842337486552943432, 6.291896048439078162688479880585, 7.312219771984035817540708561788, 7.98904150372883923623736158595, 9.680293532266764159335009712325, 10.55998385115079569154039195736, 11.48952059113813013247117037067, 12.29161793377108912145286978486, 12.7231497024683895039921213431, 13.95222070914046871197162977628, 14.479467880638080284452946898830, 15.3674981095765350679617652159, 16.41038804832817677081591553182, 17.10998814833646084514114098919, 18.04962668179582365672387172037, 18.713171957904508923325782510198, 19.61076299114828068287719034520, 20.95233585279872663493560786685, 21.599736249614515159099556453251, 22.430305598013631612527818168199, 22.61212358343229550366601319509

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