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Properties

Label 1-1792-1792.1147-r0-0-0
Degree $1$
Conductor $1792$
Sign $0.975 + 0.219i$
Analytic cond. $8.32201$
Root an. cond. $8.32201$
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  + (0.956 + 0.290i)3-s + (0.995 + 0.0980i)5-s + (0.831 + 0.555i)9-s + (−0.471 + 0.881i)11-s + (−0.0980 − 0.995i)13-s + (0.923 + 0.382i)15-s + (0.923 − 0.382i)17-s + (0.634 − 0.773i)19-s + (−0.195 − 0.980i)23-s + (0.980 + 0.195i)25-s + (0.634 + 0.773i)27-s + (−0.881 + 0.471i)29-s + (0.707 + 0.707i)31-s + (−0.707 + 0.707i)33-s + (0.773 − 0.634i)37-s + ⋯
L(s)  = 1  + (0.956 + 0.290i)3-s + (0.995 + 0.0980i)5-s + (0.831 + 0.555i)9-s + (−0.471 + 0.881i)11-s + (−0.0980 − 0.995i)13-s + (0.923 + 0.382i)15-s + (0.923 − 0.382i)17-s + (0.634 − 0.773i)19-s + (−0.195 − 0.980i)23-s + (0.980 + 0.195i)25-s + (0.634 + 0.773i)27-s + (−0.881 + 0.471i)29-s + (0.707 + 0.707i)31-s + (−0.707 + 0.707i)33-s + (0.773 − 0.634i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.975 + 0.219i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.975 + 0.219i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1792\)    =    \(2^{8} \cdot 7\)
Sign: $0.975 + 0.219i$
Analytic conductor: \(8.32201\)
Root analytic conductor: \(8.32201\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1792} (1147, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1792,\ (0:\ ),\ 0.975 + 0.219i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.982367550 + 0.3307383329i\)
\(L(\frac12)\) \(\approx\) \(2.982367550 + 0.3307383329i\)
\(L(1)\) \(\approx\) \(1.796104823 + 0.1604460398i\)
\(L(1)\) \(\approx\) \(1.796104823 + 0.1604460398i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 + (0.956 + 0.290i)T \)
5 \( 1 + (0.995 + 0.0980i)T \)
11 \( 1 + (-0.471 + 0.881i)T \)
13 \( 1 + (-0.0980 - 0.995i)T \)
17 \( 1 + (0.923 - 0.382i)T \)
19 \( 1 + (0.634 - 0.773i)T \)
23 \( 1 + (-0.195 - 0.980i)T \)
29 \( 1 + (-0.881 + 0.471i)T \)
31 \( 1 + (0.707 + 0.707i)T \)
37 \( 1 + (0.773 - 0.634i)T \)
41 \( 1 + (-0.980 + 0.195i)T \)
43 \( 1 + (0.956 - 0.290i)T \)
47 \( 1 + (-0.382 - 0.923i)T \)
53 \( 1 + (-0.881 - 0.471i)T \)
59 \( 1 + (0.0980 - 0.995i)T \)
61 \( 1 + (-0.290 + 0.956i)T \)
67 \( 1 + (0.290 - 0.956i)T \)
71 \( 1 + (0.831 - 0.555i)T \)
73 \( 1 + (-0.555 + 0.831i)T \)
79 \( 1 + (-0.382 + 0.923i)T \)
83 \( 1 + (0.773 + 0.634i)T \)
89 \( 1 + (-0.195 + 0.980i)T \)
97 \( 1 + (0.707 + 0.707i)T \)
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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

−20.41526503405226160559952347304, −19.13025218873468135498503485795, −18.91870153658666130133308921052, −18.175588134095528987340390313869, −17.216404339426426976534743601924, −16.5355777127596774887913616575, −15.74362058843403907138375843974, −14.75598969956403072639948678981, −14.110730604374252451120544640675, −13.62094939463830376174499625086, −12.99499027860947334411532029571, −12.085225676896321720572542465139, −11.20992110458299277397170062853, −10.00922789155471419330232570142, −9.66682935980810247015635144295, −8.860856383259182507011468689328, −7.99133106556644627003064158669, −7.4016009093989480221147232057, −6.19985597510166426671116669302, −5.76822145435093109330054819584, −4.57721240523825345422879839650, −3.54215294941746045362446608178, −2.82971286824708609528336767431, −1.83927029642444354102088717092, −1.19227109128753329401524702415, 1.097475662585553380179360368468, 2.21599073090182990565914534100, 2.77104355633968267806639753051, 3.62604241644998980593214079899, 4.9500027523483054222715660752, 5.24795700027839085568936900099, 6.5361200354930198850607621787, 7.38193065989957709096494804614, 8.05697173213118531504748195264, 9.01158645113998460604079268127, 9.73878712558578540700244931165, 10.19228599665528542662224689130, 10.93575061811187874648237259786, 12.33904269803497845023804001707, 12.90159063518215372586542749790, 13.614520818797378991655059325931, 14.356467984599239671898462323597, 14.92360154895936788403557320163, 15.686429631475446123392912853, 16.47023687119036250306029554935, 17.39163459719772187475000753931, 18.18369424054395704194362971592, 18.58173387452425368684842556067, 19.72451569901095552679425921102, 20.36948019051772068009555439572

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