Properties

Label 1-39-39.20-r0-0-0
Degree $1$
Conductor $39$
Sign $0.999 + 0.0257i$
Analytic cond. $0.181115$
Root an. cond. $0.181115$
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  + (−0.866 + 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)  = 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

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

Invariants

Degree: \(1\)
Conductor: \(39\)    =    \(3 \cdot 13\)
Sign: $0.999 + 0.0257i$
Analytic conductor: \(0.181115\)
Root analytic conductor: \(0.181115\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{39} (20, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 39,\ (0:\ ),\ 0.999 + 0.0257i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6007915962 + 0.007747659922i\)
\(L(\frac12)\) \(\approx\) \(0.6007915962 + 0.007747659922i\)
\(L(1)\) \(\approx\) \(0.7373464296 + 0.03723171633i\)
\(L(1)\) \(\approx\) \(0.7373464296 + 0.03723171633i\)

Euler product

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

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

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