↑

Properties

Label 1-2667-2667.2348-r0-0-0
Degree $1$
Conductor $2667$
Sign $-0.347 + 0.937i$
Analytic cond. $12.3854$
Root an. cond. $12.3854$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.0747 + 0.997i)2-s + (−0.988 − 0.149i)4-s + (0.365 − 0.930i)5-s + (0.222 − 0.974i)8-s + (0.900 + 0.433i)10-s + (0.583 − 0.811i)11-s + (−0.980 − 0.198i)13-s + (0.955 + 0.294i)16-s + (−0.0249 + 0.999i)17-s − 19-s + (−0.5 + 0.866i)20-s + (0.766 + 0.642i)22-s + (0.583 + 0.811i)23-s + (−0.733 − 0.680i)25-s + (0.270 − 0.962i)26-s + ⋯
L(s)  = 1  + (−0.0747 + 0.997i)2-s + (−0.988 − 0.149i)4-s + (0.365 − 0.930i)5-s + (0.222 − 0.974i)8-s + (0.900 + 0.433i)10-s + (0.583 − 0.811i)11-s + (−0.980 − 0.198i)13-s + (0.955 + 0.294i)16-s + (−0.0249 + 0.999i)17-s − 19-s + (−0.5 + 0.866i)20-s + (0.766 + 0.642i)22-s + (0.583 + 0.811i)23-s + (−0.733 − 0.680i)25-s + (0.270 − 0.962i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2667 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.347 + 0.937i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2667 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.347 + 0.937i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2667\)    =    \(3 \cdot 7 \cdot 127\)
Sign: $-0.347 + 0.937i$
Analytic conductor: \(12.3854\)
Root analytic conductor: \(12.3854\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2667} (2348, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2667,\ (0:\ ),\ -0.347 + 0.937i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5981700280 + 0.8600229170i\)
\(L(\frac12)\) \(\approx\) \(0.5981700280 + 0.8600229170i\)
\(L(1)\) \(\approx\) \(0.8290811765 + 0.3305474854i\)
\(L(1)\) \(\approx\) \(0.8290811765 + 0.3305474854i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
127 \( 1 \)
good2 \( 1 + (-0.0747 + 0.997i)T \)
5 \( 1 + (0.365 - 0.930i)T \)
11 \( 1 + (0.583 - 0.811i)T \)
13 \( 1 + (-0.980 - 0.198i)T \)
17 \( 1 + (-0.0249 + 0.999i)T \)
19 \( 1 - T \)
23 \( 1 + (0.583 + 0.811i)T \)
29 \( 1 + (-0.456 + 0.889i)T \)
31 \( 1 + (0.0249 + 0.999i)T \)
37 \( 1 + (0.766 + 0.642i)T \)
41 \( 1 + (0.878 - 0.478i)T \)
43 \( 1 + (-0.998 + 0.0498i)T \)
47 \( 1 + (-0.988 - 0.149i)T \)
53 \( 1 + (-0.698 + 0.715i)T \)
59 \( 1 + (0.766 + 0.642i)T \)
61 \( 1 + (0.733 - 0.680i)T \)
67 \( 1 + (0.921 - 0.388i)T \)
71 \( 1 + (-0.995 + 0.0995i)T \)
73 \( 1 + (-0.826 + 0.563i)T \)
79 \( 1 + (0.995 - 0.0995i)T \)
83 \( 1 + (-0.969 - 0.246i)T \)
89 \( 1 + (-0.900 + 0.433i)T \)
97 \( 1 + (-0.542 + 0.840i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−19.276845882287186810909751558065, −18.443459820714240425213209180475, −17.87020903373165261273210257897, −17.18734330758615457471690398812, −16.60588321322126055142171601849, −15.21956640224509369835861490297, −14.5796101570544938748076214783, −14.28063866632909678293595633541, −13.128901053678267640866183643281, −12.76634511902482594634409341552, −11.6006081339444707760425611619, −11.417699574872256298073910371978, −10.406491951226318049301203317257, −9.68143085506917103229468007532, −9.43893133172266364436369950840, −8.28945373948896572205031062458, −7.3790453460905450013269783913, −6.71320389106259908935709666743, −5.744947153185460426808524330015, −4.67333389960826107607922485234, −4.184965389142816107255339729152, −3.07249536168162852150347076548, −2.358805615280220756517554744772, −1.85905238199546615829005582540, −0.39609200025936222841971541240, 0.94623945412811656015236462378, 1.78421160699256003167436742457, 3.2306718445330624699872389334, 4.12337709876289522995019302740, 4.90150581314043095996742946211, 5.55328632852507280000673701986, 6.27935735113419139765092223481, 7.01563990848847377841319634865, 8.02232082407943826213799158768, 8.5739940588449601777922213971, 9.18827325476996684089890817310, 9.90225865297224342585662753677, 10.73875239465138753454170202710, 11.816947254434400658732240198497, 12.8674181189775329197429457345, 12.95469189114119807632353307802, 14.02111320153874923286763259301, 14.62581968700092899623541245823, 15.2704001298780076068054354634, 16.18656122517847482175498791745, 16.75229582950110675542445086511, 17.256644079658266944099242064120, 17.74625785782223169586143385963, 18.82787453975373388943725093676, 19.47960004499387443487631520517

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