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Properties

Label 1-1339-1339.1207-r0-0-0
Degree $1$
Conductor $1339$
Sign $-0.859 - 0.511i$
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  + (−0.417 − 0.908i)2-s + (−0.213 − 0.976i)3-s + (−0.650 + 0.759i)4-s + (−0.833 − 0.552i)5-s + (−0.798 + 0.602i)6-s + (0.243 − 0.969i)7-s + (0.961 + 0.273i)8-s + (−0.908 + 0.417i)9-s + (−0.153 + 0.988i)10-s + (0.995 + 0.0922i)11-s + (0.881 + 0.473i)12-s + (−0.982 + 0.183i)14-s + (−0.361 + 0.932i)15-s + (−0.153 − 0.988i)16-s + (0.389 + 0.920i)17-s + (0.759 + 0.650i)18-s + ⋯
L(s)  = 1  + (−0.417 − 0.908i)2-s + (−0.213 − 0.976i)3-s + (−0.650 + 0.759i)4-s + (−0.833 − 0.552i)5-s + (−0.798 + 0.602i)6-s + (0.243 − 0.969i)7-s + (0.961 + 0.273i)8-s + (−0.908 + 0.417i)9-s + (−0.153 + 0.988i)10-s + (0.995 + 0.0922i)11-s + (0.881 + 0.473i)12-s + (−0.982 + 0.183i)14-s + (−0.361 + 0.932i)15-s + (−0.153 − 0.988i)16-s + (0.389 + 0.920i)17-s + (0.759 + 0.650i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1339 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.859 - 0.511i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1339 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.859 - 0.511i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1339\)    =    \(13 \cdot 103\)
Sign: $-0.859 - 0.511i$
Analytic conductor: \(6.21828\)
Root analytic conductor: \(6.21828\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1339} (1207, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1339,\ (0:\ ),\ -0.859 - 0.511i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2558975000 - 0.9302474800i\)
\(L(\frac12)\) \(\approx\) \(0.2558975000 - 0.9302474800i\)
\(L(1)\) \(\approx\) \(0.5018252535 - 0.5601455228i\)
\(L(1)\) \(\approx\) \(0.5018252535 - 0.5601455228i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
103 \( 1 \)
good2 \( 1 + (-0.417 - 0.908i)T \)
3 \( 1 + (-0.213 - 0.976i)T \)
5 \( 1 + (-0.833 - 0.552i)T \)
7 \( 1 + (0.243 - 0.969i)T \)
11 \( 1 + (0.995 + 0.0922i)T \)
17 \( 1 + (0.389 + 0.920i)T \)
19 \( 1 + (-0.673 + 0.739i)T \)
23 \( 1 + (0.908 + 0.417i)T \)
29 \( 1 + (0.552 - 0.833i)T \)
31 \( 1 + (0.361 - 0.932i)T \)
37 \( 1 + (0.999 - 0.0307i)T \)
41 \( 1 + (-0.0615 - 0.998i)T \)
43 \( 1 + (-0.850 - 0.526i)T \)
47 \( 1 + (0.866 - 0.5i)T \)
53 \( 1 + (-0.213 + 0.976i)T \)
59 \( 1 + (0.961 + 0.273i)T \)
61 \( 1 + (0.992 + 0.122i)T \)
67 \( 1 + (0.961 + 0.273i)T \)
71 \( 1 + (0.833 - 0.552i)T \)
73 \( 1 + (0.895 + 0.445i)T \)
79 \( 1 + (0.445 + 0.895i)T \)
83 \( 1 + (-0.243 + 0.969i)T \)
89 \( 1 + (0.943 + 0.332i)T \)
97 \( 1 + (0.798 - 0.602i)T \)
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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

−21.580764495465422546451117734474, −20.29581241671312903443893390746, −19.55799044923858575965963753315, −18.87024496094202069301325319246, −18.05608580326312335013843606972, −17.36058853658780769893245952057, −16.384948469604414576390283515212, −15.98148067254430899501769985052, −15.120501830998691789020044350437, −14.69686380328788669474948064, −14.14313837921356740719189420784, −12.695525533561798886638789774922, −11.609687510391920781507770734583, −11.17818723242915336602310333865, −10.20939635452566961942458280911, −9.29859811628185532190381933769, −8.74400359637716422697006991962, −8.03416620721612432378878472689, −6.767277784600958860212194736908, −6.379115788155456392468330234663, −5.08548464104009326175748461587, −4.70508711632191231475295911330, −3.5914800636792588606995525264, −2.64028584563859316061366083796, −0.85612105494488779871818906248, 0.72528380087521216311603391619, 1.28012517956977395956846922180, 2.30266827297825452740047578544, 3.745869379033733276945363537756, 4.0626957957566602038878142801, 5.24920781168723610035619100172, 6.543259832777818745642046155429, 7.41856294342889392712164860687, 8.096421635653197866844336930633, 8.6416873497704093619484014182, 9.73900932161661971853889855910, 10.74390720753041366138301876154, 11.4029411362910379100296404715, 12.05898899457464608532106792551, 12.69285583083189667270384888215, 13.40442774260852354821919876085, 14.18180463061073594626641895540, 15.12677867062392716448492766594, 16.63490481457181882736996606329, 17.018030216873580525433546742526, 17.4044722792981692367171221656, 18.63390967020377712787509108724, 19.2151735077840644100484292417, 19.68374216145110552657081270020, 20.37249060384772361705364801214

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