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Properties

Label 1-1649-1649.862-r1-0-0
Degree $1$
Conductor $1649$
Sign $0.737 - 0.675i$
Analytic cond. $177.209$
Root an. cond. $177.209$
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 + (−0.991 − 0.130i)3-s + (−0.5 − 0.866i)4-s + (−0.965 + 0.258i)5-s + (−0.608 + 0.793i)6-s + (0.258 + 0.965i)7-s − 8-s + (0.965 + 0.258i)9-s + (−0.258 + 0.965i)10-s + (−0.130 − 0.991i)11-s + (0.382 + 0.923i)12-s + (0.608 + 0.793i)13-s + (0.965 + 0.258i)14-s + (0.991 − 0.130i)15-s + (−0.5 + 0.866i)16-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)2-s + (−0.991 − 0.130i)3-s + (−0.5 − 0.866i)4-s + (−0.965 + 0.258i)5-s + (−0.608 + 0.793i)6-s + (0.258 + 0.965i)7-s − 8-s + (0.965 + 0.258i)9-s + (−0.258 + 0.965i)10-s + (−0.130 − 0.991i)11-s + (0.382 + 0.923i)12-s + (0.608 + 0.793i)13-s + (0.965 + 0.258i)14-s + (0.991 − 0.130i)15-s + (−0.5 + 0.866i)16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1649 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.737 - 0.675i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1649 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.737 - 0.675i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1649\)    =    \(17 \cdot 97\)
Sign: $0.737 - 0.675i$
Analytic conductor: \(177.209\)
Root analytic conductor: \(177.209\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1649} (862, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1649,\ (1:\ ),\ 0.737 - 0.675i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6590524078 - 0.2562369644i\)
\(L(\frac12)\) \(\approx\) \(0.6590524078 - 0.2562369644i\)
\(L(1)\) \(\approx\) \(0.6219435790 - 0.2966221838i\)
\(L(1)\) \(\approx\) \(0.6219435790 - 0.2966221838i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 \)
97 \( 1 \)
good2 \( 1 + (0.5 - 0.866i)T \)
3 \( 1 + (-0.991 - 0.130i)T \)
5 \( 1 + (-0.965 + 0.258i)T \)
7 \( 1 + (0.258 + 0.965i)T \)
11 \( 1 + (-0.130 - 0.991i)T \)
13 \( 1 + (0.608 + 0.793i)T \)
19 \( 1 + (-0.923 + 0.382i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
29 \( 1 + (-0.258 - 0.965i)T \)
31 \( 1 + (-0.991 - 0.130i)T \)
37 \( 1 + (0.5 - 0.866i)T \)
41 \( 1 + (-0.866 - 0.5i)T \)
43 \( 1 + (0.258 + 0.965i)T \)
47 \( 1 - T \)
53 \( 1 + (-0.5 + 0.866i)T \)
59 \( 1 + (-0.608 - 0.793i)T \)
61 \( 1 + (0.130 - 0.991i)T \)
67 \( 1 + (0.923 + 0.382i)T \)
71 \( 1 + (0.5 + 0.866i)T \)
73 \( 1 + (0.608 + 0.793i)T \)
79 \( 1 + (-0.923 - 0.382i)T \)
83 \( 1 + (-0.793 - 0.608i)T \)
89 \( 1 + (0.707 + 0.707i)T \)
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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

−20.40829447452979332004946707506, −19.73108446227100163475278474540, −18.38157321530060081179693310775, −17.91845388886345394605648191167, −17.0725912417088071117363970076, −16.632138805393349427985400878536, −15.791144006643568102570053286, −15.26372366230123284177286105449, −14.61241033603658843067286504056, −13.32356218989347208846890521551, −12.89211552001241372570356708656, −12.13958561880401071171641754828, −11.31955911326531244761499236653, −10.62607091602659239155440404820, −9.66785909339144374976645158554, −8.51224746466924713391803261204, −7.70348002912715654207553128189, −7.13425110580326205693965459126, −6.44554894385729609444396961867, −5.30878842268377443847466655973, −4.760578546061563319769437579469, −4.00825653253151894675389978321, −3.38949191446906729777320909959, −1.50839572053349247662569169800, −0.301186779305241668323390968776, 0.41257129178731652361368870720, 1.61730157775840466845678936479, 2.51034308083776096414871768918, 3.707832990478484692390283208825, 4.281839975884541435925406154100, 5.20970481624025962524592562138, 6.08730957763859104299337733054, 6.52846893465678429791117616864, 7.9710518211682108030626267744, 8.67077200861799397019053525064, 9.65239299429975232365226753666, 10.79267810488728227878801611218, 11.1754145191182297734106943744, 11.69010257947605373803811182006, 12.47029463882962653162458361192, 12.98794403372979845219448101669, 14.13964432676572934256558911488, 14.80290943319860153957043395740, 15.7504813432415832627286068496, 16.19702047937566641665597050400, 17.24092842916624923466046857278, 18.449496398612637758819182772640, 18.64994144981852367672473921090, 19.119088856913586042361393323655, 20.15712394704407359173144493968

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