Properties

Degree 1
Conductor $ 3 \cdot 13 $
Sign $0.999 + 0.0257i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.866 − 0.5i)2-s + (0.5 + 0.866i)4-s + i·5-s + (0.866 − 0.5i)7-s i·8-s + (0.5 − 0.866i)10-s + (0.866 + 0.5i)11-s − 14-s + (−0.5 + 0.866i)16-s + (−0.5 − 0.866i)17-s + (−0.866 + 0.5i)19-s + (−0.866 + 0.5i)20-s + (−0.5 − 0.866i)22-s + (−0.5 + 0.866i)23-s − 25-s + ⋯
L(s,χ)  = 1  + (−0.866 − 0.5i)2-s + (0.5 + 0.866i)4-s + i·5-s + (0.866 − 0.5i)7-s i·8-s + (0.5 − 0.866i)10-s + (0.866 + 0.5i)11-s − 14-s + (−0.5 + 0.866i)16-s + (−0.5 − 0.866i)17-s + (−0.866 + 0.5i)19-s + (−0.866 + 0.5i)20-s + (−0.5 − 0.866i)22-s + (−0.5 + 0.866i)23-s − 25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.999 + 0.0257i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 39 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.999 + 0.0257i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(39\)    =    \(3 \cdot 13\)
\( \varepsilon \)  =  $0.999 + 0.0257i$
motivic weight  =  \(0\)
character  :  $\chi_{39} (2, \cdot )$
Sato-Tate  :  $\mu(12)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 39,\ (0:\ ),\ 0.999 + 0.0257i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(0.6007915962 + 0.007747659922i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(0.6007915962 + 0.007747659922i\)
\(L(\chi,1)\)  \(\approx\)  \(0.7373464296 - 0.03723171633i\)
\(L(1,\chi)\)  \(\approx\)  \(0.7373464296 - 0.03723171633i\)

Euler product

\[\begin{aligned}L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1}\end{aligned}\]
\[\begin{aligned}L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−35.23775027943390442839451173056, −34.35413054501015124172839002205, −32.96911304508465958867215093167, −32.10027999264631850145654601481, −30.43735141770946539286166501943, −28.88453453704830248171984782591, −27.92139887601988845319908009517, −27.12365200738229811366580058365, −25.54412937900981912657391780487, −24.470229684882051777085359461781, −23.810385521920063622330494983998, −21.68163582541555351381579675380, −20.326991882669876445594406256940, −19.2171752591164357579040384650, −17.71192864266900011600837923948, −16.82442895707711357503167567511, −15.5125507908930185713788349979, −14.20290583255318815161949026861, −12.21788248051765929806666929317, −10.79438277663158086969655165629, −8.94740637583644850926117514264, −8.32192734799383652576732018588, −6.34763763429878864065979309906, −4.803286825591967598580441021816, −1.63445239286509423594069645526, 2.03404711878085363810921815506, 3.95019113494697740624558412136, 6.72435957016590731609804206081, 7.9156761518115367373494124713, 9.63717734594220734388391276160, 10.90112628955612568995291181392, 11.88320536485211017265965124799, 13.89162637911162631521345651842, 15.32100379962082372319456450285, 17.11044005221097058965736739039, 17.968046897996614700574194305728, 19.21191659707203630816815231779, 20.40416734683626654258107462078, 21.65710564050586822572042397273, 22.94050059090525224787095735431, 24.738433160648302013850033256824, 25.94933613705387979202767547869, 27.05892140378821799776753531061, 27.81575157944387021091236749410, 29.52849446798318931456182997606, 30.19398449027874475178368113473, 31.2401618267578964182641640024, 33.41417568960338952002420043605, 34.10528175204730958065941129128, 35.42943521414631492805559011049

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