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Properties

Label 1-39-39.23-r1-0-0
Degree $1$
Conductor $39$
Sign $0.0128 + 0.999i$
Analytic cond. $4.19113$
Root an. cond. $4.19113$
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  + (−0.5 + 0.866i)2-s + (−0.5 − 0.866i)4-s + 5-s + (0.5 + 0.866i)7-s + 8-s + (−0.5 + 0.866i)10-s + (−0.5 + 0.866i)11-s − 14-s + (−0.5 + 0.866i)16-s + (0.5 + 0.866i)17-s + (0.5 + 0.866i)19-s + (−0.5 − 0.866i)20-s + (−0.5 − 0.866i)22-s + (0.5 − 0.866i)23-s + 25-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)2-s + (−0.5 − 0.866i)4-s + 5-s + (0.5 + 0.866i)7-s + 8-s + (−0.5 + 0.866i)10-s + (−0.5 + 0.866i)11-s − 14-s + (−0.5 + 0.866i)16-s + (0.5 + 0.866i)17-s + (0.5 + 0.866i)19-s + (−0.5 − 0.866i)20-s + (−0.5 − 0.866i)22-s + (0.5 − 0.866i)23-s + 25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0128 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0128 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(39\)    =    \(3 \cdot 13\)
Sign: $0.0128 + 0.999i$
Analytic conductor: \(4.19113\)
Root analytic conductor: \(4.19113\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{39} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 39,\ (1:\ ),\ 0.0128 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9187333956 + 0.9070267570i\)
\(L(\frac12)\) \(\approx\) \(0.9187333956 + 0.9070267570i\)
\(L(1)\) \(\approx\) \(0.8777002573 + 0.4918428369i\)
\(L(1)\) \(\approx\) \(0.8777002573 + 0.4918428369i\)

Euler product

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

−34.81412946176364821352328738257, −33.547928108528962771863399278889, −32.21577619981980030714463134938, −30.79251807338368513801682988973, −29.60241409737823183259506033967, −29.04616301919069493448632123447, −27.49979100847375867730585268824, −26.506431845375788556147305332761, −25.359748225330826742543741490070, −23.708765044419139864993265953220, −22.06917458748376347714034927810, −21.09729152994569869694033184777, −20.108446217290996805602298071239, −18.5460290501461449316082617394, −17.552342349116300763199388354048, −16.44370229235040486575902443689, −14.01350292153561793815861249016, −13.213687945727177998139203674956, −11.37454049376359253353363419458, −10.297341359147868890087193487925, −9.004966630550517714944566873989, −7.37198339413050143084043857740, −5.08025849488051124824528730656, −3.03051772381527771939117114118, −1.13661584237586133250823858123, 1.87881590272478459477225235084, 5.04817757865145753193065436620, 6.17390079629104829599285408986, 7.91492464863991371554807006731, 9.29789717866723374375552268742, 10.46851333479332851221914890761, 12.654768619306332455475972928234, 14.25664297917617296707514238540, 15.22429626527556528045924680275, 16.7785195279511758675863036186, 17.9058345873361278368378105983, 18.73180914493737153776103351270, 20.62287467979597522593597744459, 21.983855020374874777914676311237, 23.39353679924658706346235203711, 24.82657073062813873791475509252, 25.42813983454312159511488769523, 26.69022678418406615859446697366, 28.14328693319921429710718164482, 28.84555918052751054607790768823, 30.67135872836632377805446413088, 32.01302900634730591563553907790, 33.22012397115354004058026326876, 34.009948114126599040996765199280, 35.123624437194518717405555899967

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