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Properties

Label 1-3968-3968.1379-r0-0-0
Degree $1$
Conductor $3968$
Sign $0.500 - 0.866i$
Analytic cond. $18.4273$
Root an. cond. $18.4273$
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  + (−0.117 − 0.993i)3-s + (0.555 − 0.831i)5-s + (−0.972 − 0.233i)7-s + (−0.972 + 0.233i)9-s + (−0.418 + 0.908i)11-s + (−0.619 + 0.785i)13-s + (−0.891 − 0.453i)15-s + (−0.987 − 0.156i)17-s + (−0.271 − 0.962i)19-s + (−0.117 + 0.993i)21-s + (0.233 + 0.972i)23-s + (−0.382 − 0.923i)25-s + (0.346 + 0.938i)27-s + (0.872 + 0.488i)29-s + (0.951 + 0.309i)33-s + ⋯
L(s)  = 1  + (−0.117 − 0.993i)3-s + (0.555 − 0.831i)5-s + (−0.972 − 0.233i)7-s + (−0.972 + 0.233i)9-s + (−0.418 + 0.908i)11-s + (−0.619 + 0.785i)13-s + (−0.891 − 0.453i)15-s + (−0.987 − 0.156i)17-s + (−0.271 − 0.962i)19-s + (−0.117 + 0.993i)21-s + (0.233 + 0.972i)23-s + (−0.382 − 0.923i)25-s + (0.346 + 0.938i)27-s + (0.872 + 0.488i)29-s + (0.951 + 0.309i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3968 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.500 - 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3968 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.500 - 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3968\)    =    \(2^{7} \cdot 31\)
Sign: $0.500 - 0.866i$
Analytic conductor: \(18.4273\)
Root analytic conductor: \(18.4273\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3968} (1379, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3968,\ (0:\ ),\ 0.500 - 0.866i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8867435921 - 0.5119563137i\)
\(L(\frac12)\) \(\approx\) \(0.8867435921 - 0.5119563137i\)
\(L(1)\) \(\approx\) \(0.7794731939 - 0.3138609406i\)
\(L(1)\) \(\approx\) \(0.7794731939 - 0.3138609406i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
31 \( 1 \)
good3 \( 1 + (-0.117 - 0.993i)T \)
5 \( 1 + (0.555 - 0.831i)T \)
7 \( 1 + (-0.972 - 0.233i)T \)
11 \( 1 + (-0.418 + 0.908i)T \)
13 \( 1 + (-0.619 + 0.785i)T \)
17 \( 1 + (-0.987 - 0.156i)T \)
19 \( 1 + (-0.271 - 0.962i)T \)
23 \( 1 + (0.233 + 0.972i)T \)
29 \( 1 + (0.872 + 0.488i)T \)
37 \( 1 + (-0.831 - 0.555i)T \)
41 \( 1 + (0.760 + 0.649i)T \)
43 \( 1 + (-0.117 + 0.993i)T \)
47 \( 1 + (0.453 - 0.891i)T \)
53 \( 1 + (0.734 + 0.678i)T \)
59 \( 1 + (0.962 + 0.271i)T \)
61 \( 1 + (-0.980 - 0.195i)T \)
67 \( 1 + (-0.980 - 0.195i)T \)
71 \( 1 + (0.522 - 0.852i)T \)
73 \( 1 + (-0.522 - 0.852i)T \)
79 \( 1 + (-0.156 + 0.987i)T \)
83 \( 1 + (-0.271 - 0.962i)T \)
89 \( 1 + (-0.233 + 0.972i)T \)
97 \( 1 + (-0.587 + 0.809i)T \)
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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

−18.695671575671981479454621022003, −17.82684993714893136979424040272, −17.185072090736839655006238657874, −16.56322849199958198758483626418, −15.709261212286560211448449479401, −15.421140543349502130622067383379, −14.549600343516736059592202398652, −13.98910005598280159660719575634, −13.203489911457690198039782572531, −12.461428823824810169312302136862, −11.59921949857769274382847198286, −10.70499019287294100965571176510, −10.31303533902540341835490116552, −9.895753750236922850791575671690, −8.88234144798464252346516631598, −8.44618275419753366646900249420, −7.28977530140766471001151470858, −6.391219667351900139701526167183, −5.91910037629992967187175042259, −5.285839575649322857210990721236, −4.25246153086562067607005957682, −3.432637995247969420797378313835, −2.800658851223534438010583817742, −2.276043918670343723760373682961, −0.50360870367873703394745098172, 0.54529762968361599073643369764, 1.60325954436874141226401177149, 2.29061899253945950344150765083, 2.96309318428159085479409253687, 4.31064840313895654485665937687, 4.89725272426255534784212622210, 5.73159986008239373376472820880, 6.594624969966173872373723278576, 7.01975519423352179769106854822, 7.71673819822360102140495247755, 8.87692701172341151040305853414, 9.157833174442272448204420745852, 9.97294214329467812300147496170, 10.81915396172726112459780805560, 11.77209273593124794450817042100, 12.37636149793642914335349873705, 12.93713347384966569775075070894, 13.47865873073342600232365041008, 13.93557245957829521639196527300, 14.99640309887881609221669785486, 15.77597494495050352337364903770, 16.51628560712902020741258801724, 17.11654085021750086597873802106, 17.786163143952318755528222217626, 18.1217272894186046320456068459

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