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Properties

Label 1-1339-1339.102-r0-0-0
Degree $1$
Conductor $1339$
Sign $-0.0386 + 0.999i$
Analytic cond. $6.21828$
Root an. cond. $6.21828$
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)3-s + (0.5 + 0.866i)4-s − i·5-s + (−0.866 + 0.5i)6-s + (−0.866 + 0.5i)7-s − i·8-s + (−0.5 − 0.866i)9-s + (−0.5 + 0.866i)10-s + (−0.866 − 0.5i)11-s + 12-s + 14-s + (−0.866 − 0.5i)15-s + (−0.5 + 0.866i)16-s + (0.5 + 0.866i)17-s + i·18-s + ⋯
L(s)  = 1  + (−0.866 − 0.5i)2-s + (0.5 − 0.866i)3-s + (0.5 + 0.866i)4-s − i·5-s + (−0.866 + 0.5i)6-s + (−0.866 + 0.5i)7-s − i·8-s + (−0.5 − 0.866i)9-s + (−0.5 + 0.866i)10-s + (−0.866 − 0.5i)11-s + 12-s + 14-s + (−0.866 − 0.5i)15-s + (−0.5 + 0.866i)16-s + (0.5 + 0.866i)17-s + i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1339 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0386 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1339 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0386 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1339\)    =    \(13 \cdot 103\)
Sign: $-0.0386 + 0.999i$
Analytic conductor: \(6.21828\)
Root analytic conductor: \(6.21828\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1339} (102, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1339,\ (0:\ ),\ -0.0386 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1621431190 - 0.1685281414i\)
\(L(\frac12)\) \(\approx\) \(-0.1621431190 - 0.1685281414i\)
\(L(1)\) \(\approx\) \(0.4587720768 - 0.3942809572i\)
\(L(1)\) \(\approx\) \(0.4587720768 - 0.3942809572i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
103 \( 1 \)
good2 \( 1 + (-0.866 - 0.5i)T \)
3 \( 1 + (0.5 - 0.866i)T \)
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

−21.280747323822022300943719893315, −20.535302404119353206489885489, −19.87967498956484201349960285293, −19.11182527499670221026444668979, −18.521667234865556457663439593636, −17.69378294354272100710271728369, −16.763377657116081586977291021653, −16.07974766172709975017709529571, −15.47167642844148307868241758262, −14.91594801009229335312661218330, −13.93228504804608234296638676311, −13.51127852467923120215170417501, −11.91122976785276221803176866922, −11.018726309247239527348328074939, −10.23192281318765410653047781648, −9.85836560838132722677678664715, −9.2403599776158790914309808061, −7.952732101623891144717596513815, −7.47433426094325018088035926327, −6.67734586171109035923784134692, −5.61283901939464663269556383845, −4.82803798613489816375269511312, −3.32502607018119820979209677512, −2.98314806330232227203252674257, −1.74115017207201330689316938834, 0.11705399442844157458203796165, 1.148475323834352902657947085384, 2.12708852641497669732192524143, 3.00802842050806707413164362065, 3.72243667871718513222785053820, 5.28734650115524577137900151617, 6.174840443846414521781695417091, 7.15401082564308636744309773173, 7.97202961695763926256067016104, 8.65455181142581823215938994953, 9.18037811970175328859490310421, 10.00132985380520337705407665278, 11.0531702359229921405714686696, 12.04700098654878645588565048869, 12.66167507861405243877384535249, 13.02947450335927182240037340774, 13.890367228872797046184196710718, 15.27400908664591505434507542078, 15.90952257314319275303718354135, 16.749491856565436602064349049376, 17.35923144404573042140272345607, 18.41654893481191769757832101671, 18.815969487504359732587748450881, 19.47198474785435605378029737369, 20.284092706276736072010706222327

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