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Properties

Label 1-1339-1339.355-r0-0-0
Degree $1$
Conductor $1339$
Sign $0.970 + 0.242i$
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  − 2-s + (−0.5 − 0.866i)3-s + 4-s + (0.5 + 0.866i)5-s + (0.5 + 0.866i)6-s − 7-s − 8-s + (−0.5 + 0.866i)9-s + (−0.5 − 0.866i)10-s + (0.5 − 0.866i)11-s + (−0.5 − 0.866i)12-s + 14-s + (0.5 − 0.866i)15-s + 16-s + 17-s + (0.5 − 0.866i)18-s + ⋯
L(s)  = 1  − 2-s + (−0.5 − 0.866i)3-s + 4-s + (0.5 + 0.866i)5-s + (0.5 + 0.866i)6-s − 7-s − 8-s + (−0.5 + 0.866i)9-s + (−0.5 − 0.866i)10-s + (0.5 − 0.866i)11-s + (−0.5 − 0.866i)12-s + 14-s + (0.5 − 0.866i)15-s + 16-s + 17-s + (0.5 − 0.866i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1339 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.970 + 0.242i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1339 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.970 + 0.242i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1339\)    =    \(13 \cdot 103\)
Sign: $0.970 + 0.242i$
Analytic conductor: \(6.21828\)
Root analytic conductor: \(6.21828\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1339} (355, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1339,\ (0:\ ),\ 0.970 + 0.242i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7168958172 + 0.08808863440i\)
\(L(\frac12)\) \(\approx\) \(0.7168958172 + 0.08808863440i\)
\(L(1)\) \(\approx\) \(0.6089104368 - 0.04436462080i\)
\(L(1)\) \(\approx\) \(0.6089104368 - 0.04436462080i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
103 \( 1 \)
good2 \( 1 - T \)
3 \( 1 + (-0.5 - 0.866i)T \)
5 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 - T \)
11 \( 1 + (0.5 - 0.866i)T \)
17 \( 1 + T \)
19 \( 1 + (0.5 + 0.866i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
29 \( 1 + T \)
31 \( 1 - T \)
37 \( 1 + (0.5 + 0.866i)T \)
41 \( 1 - T \)
43 \( 1 + (-0.5 - 0.866i)T \)
47 \( 1 + (0.5 + 0.866i)T \)
53 \( 1 + (-0.5 - 0.866i)T \)
59 \( 1 + (0.5 + 0.866i)T \)
61 \( 1 + (-0.5 + 0.866i)T \)
67 \( 1 + (0.5 - 0.866i)T \)
71 \( 1 - T \)
73 \( 1 - T \)
79 \( 1 + T \)
83 \( 1 + (0.5 - 0.866i)T \)
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

−20.6846572716025323284570878424, −20.040498041489785389305447401027, −19.63455126812881469579047622602, −18.39599420606576236426419124986, −17.62012700296174896776782415793, −17.09534339110711135537997253420, −16.3694418127921489813455454569, −15.91232271801741434705141822931, −15.15116058970473596058253590907, −14.14244965361415216792099077149, −12.8990512125312348264290395083, −12.16728491800852078250596886006, −11.58570005331762743420852826068, −10.41612503041661025819108384391, −9.76350848812313697893142660130, −9.42498753760389778326602306860, −8.70773431560649543832308852552, −7.50642752726914738940563272920, −6.576689826561771357572691189247, −5.80763685723596496680850390675, −5.03257885532991675704794423431, −3.856671236460026985746114147897, −2.94355299268717800658726070041, −1.662701802067004846717286954602, −0.57697787425072492324957187602, 0.83312219634445668910739595527, 1.81282000060797499252803283238, 2.86206229574784546368875250564, 3.471791260071534454910706524898, 5.54090415914318122067969788734, 6.17315941633253056526663203040, 6.64052938836412431223339507727, 7.47853365887140027286381746460, 8.28434159998722768639825375200, 9.268290909155789005955913047, 10.25038328093796234794928889863, 10.55774486735713882191810632073, 11.74857228696620654509955871342, 12.11792487051978997141440372254, 13.20732702342292983307389054904, 14.0636637093923807760210631470, 14.75780857282433027986294670133, 16.06386418050451838262419271317, 16.56278947051805036830978289061, 17.1674533111801426324179851738, 18.15529728285056925673996992828, 18.65093621850171227241598072127, 19.07914458851228472882588925660, 19.810862383242555619525675839040, 20.761083995108105809045533797993

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