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Properties

Label 1-1343-1343.802-r0-0-0
Degree $1$
Conductor $1343$
Sign $-0.991 - 0.130i$
Analytic cond. $6.23686$
Root an. cond. $6.23686$
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  + (−0.517 + 0.855i)2-s + (0.437 + 0.899i)3-s + (−0.464 − 0.885i)4-s + (−0.977 + 0.209i)5-s + (−0.995 − 0.0904i)6-s + (0.326 − 0.945i)7-s + (0.998 + 0.0603i)8-s + (−0.616 + 0.787i)9-s + (0.326 − 0.945i)10-s + (−0.209 + 0.977i)11-s + (0.592 − 0.805i)12-s + (0.464 + 0.885i)13-s + (0.640 + 0.768i)14-s + (−0.616 − 0.787i)15-s + (−0.568 + 0.822i)16-s + ⋯
L(s)  = 1  + (−0.517 + 0.855i)2-s + (0.437 + 0.899i)3-s + (−0.464 − 0.885i)4-s + (−0.977 + 0.209i)5-s + (−0.995 − 0.0904i)6-s + (0.326 − 0.945i)7-s + (0.998 + 0.0603i)8-s + (−0.616 + 0.787i)9-s + (0.326 − 0.945i)10-s + (−0.209 + 0.977i)11-s + (0.592 − 0.805i)12-s + (0.464 + 0.885i)13-s + (0.640 + 0.768i)14-s + (−0.616 − 0.787i)15-s + (−0.568 + 0.822i)16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1343 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.991 - 0.130i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1343 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.991 - 0.130i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1343\)    =    \(17 \cdot 79\)
Sign: $-0.991 - 0.130i$
Analytic conductor: \(6.23686\)
Root analytic conductor: \(6.23686\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1343} (802, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1343,\ (0:\ ),\ -0.991 - 0.130i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.05516844947 + 0.8418570658i\)
\(L(\frac12)\) \(\approx\) \(-0.05516844947 + 0.8418570658i\)
\(L(1)\) \(\approx\) \(0.5402623128 + 0.5380484518i\)
\(L(1)\) \(\approx\) \(0.5402623128 + 0.5380484518i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 \)
79 \( 1 \)
good2 \( 1 + (-0.517 + 0.855i)T \)
3 \( 1 + (0.437 + 0.899i)T \)
5 \( 1 + (-0.977 + 0.209i)T \)
7 \( 1 + (0.326 - 0.945i)T \)
11 \( 1 + (-0.209 + 0.977i)T \)
13 \( 1 + (0.464 + 0.885i)T \)
19 \( 1 + (-0.911 - 0.410i)T \)
23 \( 1 + (0.923 - 0.382i)T \)
29 \( 1 + (0.871 + 0.491i)T \)
31 \( 1 + (0.988 + 0.150i)T \)
37 \( 1 + (-0.0302 + 0.999i)T \)
41 \( 1 + (-0.542 + 0.839i)T \)
43 \( 1 + (0.180 - 0.983i)T \)
47 \( 1 + (0.935 + 0.354i)T \)
53 \( 1 + (0.0603 - 0.998i)T \)
59 \( 1 + (0.297 + 0.954i)T \)
61 \( 1 + (-0.728 + 0.685i)T \)
67 \( 1 + (0.970 + 0.239i)T \)
71 \( 1 + (0.945 - 0.326i)T \)
73 \( 1 + (-0.768 + 0.640i)T \)
83 \( 1 + (-0.297 + 0.954i)T \)
89 \( 1 + (-0.663 + 0.748i)T \)
97 \( 1 + (-0.999 - 0.0302i)T \)
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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

−20.38076962332212378103301320040, −19.54379145999790212526496560297, −18.92365504525593578670934361349, −18.680382750396171754235131633375, −17.69463782587423160705932204774, −17.03411978306295574447929294907, −15.86437696428269870408462435229, −15.255869364778723519032375182336, −14.19043805839925709100189476842, −13.30724443956351116510383290061, −12.58146299176988168667537383822, −12.087352498172492070063527955580, −11.268052648189331571789202984020, −10.694565720551153233407885120059, −9.310898531084120445682383538590, −8.44140967614058301216616922707, −8.31152457881898539591589545062, −7.49189230409752001363055997668, −6.30359131828106289535305081871, −5.26317706315309709230996528318, −4.02418628042494824601322560120, −3.12144524764408642733510372477, −2.556665350020031213609362854956, −1.35444065393007308445086308064, −0.4383902489438954439690801936, 1.18969307072916487759690370574, 2.602384269707467507460002196593, 3.95918229791867429543544689447, 4.463007115328953349429533614471, 5.04367658821178804208502362574, 6.67726813161692146671239530815, 7.05071509829950025232319828430, 8.1485787780370012762651182738, 8.5399139273089417051420417838, 9.518818350731289108444320531509, 10.44224071646590134733742190458, 10.809557433768382058469623356611, 11.79193638862554948034260816916, 13.170478563596777142882484967041, 13.995968782917107453145119603478, 14.69407740002065642747822018935, 15.29279383822141402975500922512, 15.86171421089336994135773582593, 16.75351356506375546775729702415, 17.15403500312658263741240479182, 18.22436703544148640247168967486, 19.15043414800287758508838934642, 19.66873894751815811255171529457, 20.41951994657277317788512489626, 21.0919142982546946502537803803

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