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Properties

Label 1-111-111.86-r1-0-0
Degree $1$
Conductor $111$
Sign $0.806 + 0.590i$
Analytic cond. $11.9286$
Root an. cond. $11.9286$
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  + (−0.173 + 0.984i)2-s + (−0.939 − 0.342i)4-s + (−0.766 + 0.642i)5-s + (0.766 − 0.642i)7-s + (0.5 − 0.866i)8-s + (−0.5 − 0.866i)10-s + (0.5 − 0.866i)11-s + (−0.939 − 0.342i)13-s + (0.5 + 0.866i)14-s + (0.766 + 0.642i)16-s + (0.939 − 0.342i)17-s + (0.173 + 0.984i)19-s + (0.939 − 0.342i)20-s + (0.766 + 0.642i)22-s + (0.5 + 0.866i)23-s + ⋯
L(s)  = 1  + (−0.173 + 0.984i)2-s + (−0.939 − 0.342i)4-s + (−0.766 + 0.642i)5-s + (0.766 − 0.642i)7-s + (0.5 − 0.866i)8-s + (−0.5 − 0.866i)10-s + (0.5 − 0.866i)11-s + (−0.939 − 0.342i)13-s + (0.5 + 0.866i)14-s + (0.766 + 0.642i)16-s + (0.939 − 0.342i)17-s + (0.173 + 0.984i)19-s + (0.939 − 0.342i)20-s + (0.766 + 0.642i)22-s + (0.5 + 0.866i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 111 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.806 + 0.590i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 111 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.806 + 0.590i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(111\)    =    \(3 \cdot 37\)
Sign: $0.806 + 0.590i$
Analytic conductor: \(11.9286\)
Root analytic conductor: \(11.9286\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{111} (86, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 111,\ (1:\ ),\ 0.806 + 0.590i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.246945659 + 0.4077687171i\)
\(L(\frac12)\) \(\approx\) \(1.246945659 + 0.4077687171i\)
\(L(1)\) \(\approx\) \(0.8695526414 + 0.3266812664i\)
\(L(1)\) \(\approx\) \(0.8695526414 + 0.3266812664i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
37 \( 1 \)
good2 \( 1 + (-0.173 + 0.984i)T \)
5 \( 1 + (-0.766 + 0.642i)T \)
7 \( 1 + (0.766 - 0.642i)T \)
11 \( 1 + (0.5 - 0.866i)T \)
13 \( 1 + (-0.939 - 0.342i)T \)
17 \( 1 + (0.939 - 0.342i)T \)
19 \( 1 + (0.173 + 0.984i)T \)
23 \( 1 + (0.5 + 0.866i)T \)
29 \( 1 + (0.5 - 0.866i)T \)
31 \( 1 + T \)
41 \( 1 + (0.939 + 0.342i)T \)
43 \( 1 + T \)
47 \( 1 + (0.5 + 0.866i)T \)
53 \( 1 + (-0.766 - 0.642i)T \)
59 \( 1 + (-0.766 - 0.642i)T \)
61 \( 1 + (-0.939 - 0.342i)T \)
67 \( 1 + (0.766 - 0.642i)T \)
71 \( 1 + (-0.173 - 0.984i)T \)
73 \( 1 + T \)
79 \( 1 + (0.766 - 0.642i)T \)
83 \( 1 + (0.939 - 0.342i)T \)
89 \( 1 + (-0.766 - 0.642i)T \)
97 \( 1 + (-0.5 - 0.866i)T \)
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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

−28.88820851480900210152928810714, −28.00116766906652428910313081064, −27.49913777775850614667250928479, −26.38843404622870741282364750826, −24.924411129132249113801582453904, −23.86889313025629248203277660048, −22.775114473761695950658711385570, −21.66817884710833328072262546489, −20.7315461354878300921792087079, −19.78771703416484004627787698564, −18.956125717401562001171344562784, −17.69514810629689077517121601644, −16.81645984302912814627675653018, −15.209088265803374692586846194244, −14.20188611684642864690269743777, −12.43667123125268657385023637560, −12.15221837159800079117088531945, −10.9547868438079024141360668051, −9.49264428695125261526517109662, −8.57864307663699423674798610525, −7.41912945621212526699770989175, −5.06815519799252899302742517368, −4.310364917646290221278684673440, −2.58470593591586848351120110192, −1.08995292441564824993966806489, 0.804829059069862917684747266764, 3.44848836231162585988634052660, 4.714166405152985462874047554734, 6.1445247283357375197947382137, 7.55450075523829046912879693567, 7.997371403264015663109148353916, 9.66787442557344056584784475046, 10.87256746356887889477039756154, 12.146732703138407961531646926986, 13.93692170225886652651703787878, 14.47761760662906571264844669592, 15.58329130115526249417045680207, 16.723558912848323981966665051566, 17.5979171390154961126495950305, 18.8526108610111985713855218998, 19.56969358063483828024142114150, 21.190356265782035131908153664919, 22.52328326088000615678335731996, 23.25170642357203097622467460721, 24.23695803730445865276015707827, 25.09268104171895346570503894160, 26.47240696376025228868247815204, 27.160242426633400970794728002108, 27.61868468898016010825085681876, 29.44401899658420892854259620055

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