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Properties

Label 1-417-417.74-r0-0-0
Degree $1$
Conductor $417$
Sign $-0.0681 + 0.997i$
Analytic cond. $1.93653$
Root an. cond. $1.93653$
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  + (0.854 + 0.519i)2-s + (0.460 + 0.887i)4-s + (0.990 − 0.136i)5-s + (−0.576 + 0.816i)7-s + (−0.0682 + 0.997i)8-s + (0.917 + 0.398i)10-s + (0.775 + 0.631i)11-s + (−0.917 − 0.398i)13-s + (−0.917 + 0.398i)14-s + (−0.576 + 0.816i)16-s + (−0.334 + 0.942i)17-s + (0.334 − 0.942i)19-s + (0.576 + 0.816i)20-s + (0.334 + 0.942i)22-s + (−0.576 + 0.816i)23-s + ⋯
L(s)  = 1  + (0.854 + 0.519i)2-s + (0.460 + 0.887i)4-s + (0.990 − 0.136i)5-s + (−0.576 + 0.816i)7-s + (−0.0682 + 0.997i)8-s + (0.917 + 0.398i)10-s + (0.775 + 0.631i)11-s + (−0.917 − 0.398i)13-s + (−0.917 + 0.398i)14-s + (−0.576 + 0.816i)16-s + (−0.334 + 0.942i)17-s + (0.334 − 0.942i)19-s + (0.576 + 0.816i)20-s + (0.334 + 0.942i)22-s + (−0.576 + 0.816i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 417 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0681 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 417 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0681 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(417\)    =    \(3 \cdot 139\)
Sign: $-0.0681 + 0.997i$
Analytic conductor: \(1.93653\)
Root analytic conductor: \(1.93653\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{417} (74, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 417,\ (0:\ ),\ -0.0681 + 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.607952638 + 1.721622774i\)
\(L(\frac12)\) \(\approx\) \(1.607952638 + 1.721622774i\)
\(L(1)\) \(\approx\) \(1.593483116 + 0.8787177434i\)
\(L(1)\) \(\approx\) \(1.593483116 + 0.8787177434i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
139 \( 1 \)
good2 \( 1 + (0.854 + 0.519i)T \)
5 \( 1 + (0.990 - 0.136i)T \)
7 \( 1 + (-0.576 + 0.816i)T \)
11 \( 1 + (0.775 + 0.631i)T \)
13 \( 1 + (-0.917 - 0.398i)T \)
17 \( 1 + (-0.334 + 0.942i)T \)
19 \( 1 + (0.334 - 0.942i)T \)
23 \( 1 + (-0.576 + 0.816i)T \)
29 \( 1 + (-0.460 - 0.887i)T \)
31 \( 1 + (0.682 - 0.730i)T \)
37 \( 1 + (0.962 - 0.269i)T \)
41 \( 1 + (-0.203 + 0.979i)T \)
43 \( 1 - T \)
47 \( 1 + (0.990 + 0.136i)T \)
53 \( 1 + (0.854 + 0.519i)T \)
59 \( 1 + (0.460 - 0.887i)T \)
61 \( 1 + (-0.203 - 0.979i)T \)
67 \( 1 + (-0.775 + 0.631i)T \)
71 \( 1 + (0.990 - 0.136i)T \)
73 \( 1 + (0.0682 - 0.997i)T \)
79 \( 1 + (0.203 + 0.979i)T \)
83 \( 1 + (0.775 + 0.631i)T \)
89 \( 1 + (-0.962 - 0.269i)T \)
97 \( 1 - T \)
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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

−24.03472781228477253975415524552, −22.8262664988424824496018853204, −22.283178236387102260006033681442, −21.61082148888713549412095983001, −20.57497405205460624611497475637, −19.94691646574085050297724225576, −19.00578188260957480779155959396, −18.09528672766379583375654397951, −16.74619217103006662445710744865, −16.33053672121325249270951116513, −14.813374632477193083166195858644, −14.02164431409298108198054496336, −13.64218042583960878107906166962, −12.527016031194055644450268820151, −11.6934207854884449120913636843, −10.50899967509136660732426520479, −9.92246394036525345920962732832, −9.02398463962293209139529648946, −7.14251576131444191465484592235, −6.491417788387186543580392617179, −5.50770941853512224435943376510, −4.41283344859110014190211875726, −3.34878388636585170459462648490, −2.322898964716270425766365991851, −1.07570991391035924414447209367, 1.98281174217934009576187331220, 2.78154831303303871027658387204, 4.15498736686514241382521315508, 5.23224948428904574874913242587, 6.05981481810695678816177819166, 6.78114324579317030171885833841, 8.00157868105793263158158774856, 9.22912898583187698635614071501, 9.88522370886973213745268140611, 11.42435536266245013604410158326, 12.3238475434424188058227467339, 13.05044461913392023935997947347, 13.8137687687993908362989921545, 15.00450060323097774887386684946, 15.32464321078192197779911682102, 16.67463587761064301605717386102, 17.31876210607051829671832165648, 18.04675137867184894191828485721, 19.517311272698441933808278327891, 20.24865773051675325605160867445, 21.456732407713950833748629023325, 22.02040745398542561916771643978, 22.454578354145356706249342794751, 23.66833886545029642584268751527, 24.71103509943027617471483772302

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