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Properties

Label 1-4031-4031.626-r0-0-0
Degree $1$
Conductor $4031$
Sign $-0.121 - 0.992i$
Analytic cond. $18.7198$
Root an. cond. $18.7198$
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  + (−0.0455 − 0.998i)2-s + (−0.956 + 0.291i)3-s + (−0.995 + 0.0909i)4-s + (0.715 − 0.699i)5-s + (0.334 + 0.942i)6-s + (−0.648 − 0.761i)7-s + (0.136 + 0.990i)8-s + (0.829 − 0.557i)9-s + (−0.730 − 0.682i)10-s + (0.313 + 0.949i)11-s + (0.926 − 0.377i)12-s + (0.974 + 0.225i)13-s + (−0.730 + 0.682i)14-s + (−0.480 + 0.877i)15-s + (0.983 − 0.181i)16-s + (0.987 − 0.158i)17-s + ⋯
L(s)  = 1  + (−0.0455 − 0.998i)2-s + (−0.956 + 0.291i)3-s + (−0.995 + 0.0909i)4-s + (0.715 − 0.699i)5-s + (0.334 + 0.942i)6-s + (−0.648 − 0.761i)7-s + (0.136 + 0.990i)8-s + (0.829 − 0.557i)9-s + (−0.730 − 0.682i)10-s + (0.313 + 0.949i)11-s + (0.926 − 0.377i)12-s + (0.974 + 0.225i)13-s + (−0.730 + 0.682i)14-s + (−0.480 + 0.877i)15-s + (0.983 − 0.181i)16-s + (0.987 − 0.158i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4031 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.121 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4031 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.121 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4031\)    =    \(29 \cdot 139\)
Sign: $-0.121 - 0.992i$
Analytic conductor: \(18.7198\)
Root analytic conductor: \(18.7198\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4031} (626, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4031,\ (0:\ ),\ -0.121 - 0.992i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8260760257 - 0.9333362269i\)
\(L(\frac12)\) \(\approx\) \(0.8260760257 - 0.9333362269i\)
\(L(1)\) \(\approx\) \(0.7199006587 - 0.4385549490i\)
\(L(1)\) \(\approx\) \(0.7199006587 - 0.4385549490i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad29 \( 1 \)
139 \( 1 \)
good2 \( 1 + (-0.0455 - 0.998i)T \)
3 \( 1 + (-0.956 + 0.291i)T \)
5 \( 1 + (0.715 - 0.699i)T \)
7 \( 1 + (-0.648 - 0.761i)T \)
11 \( 1 + (0.313 + 0.949i)T \)
13 \( 1 + (0.974 + 0.225i)T \)
17 \( 1 + (0.987 - 0.158i)T \)
19 \( 1 + (-0.356 + 0.934i)T \)
23 \( 1 + (0.334 - 0.942i)T \)
31 \( 1 + (-0.557 + 0.829i)T \)
37 \( 1 + (-0.480 - 0.877i)T \)
41 \( 1 + (0.993 + 0.113i)T \)
43 \( 1 + (0.866 - 0.5i)T \)
47 \( 1 + (-0.968 - 0.247i)T \)
53 \( 1 + (-0.538 - 0.842i)T \)
59 \( 1 + (0.576 - 0.816i)T \)
61 \( 1 + (-0.993 + 0.113i)T \)
67 \( 1 + (-0.746 + 0.665i)T \)
71 \( 1 + (0.247 + 0.968i)T \)
73 \( 1 + (0.789 - 0.613i)T \)
79 \( 1 + (-0.398 + 0.917i)T \)
83 \( 1 + (0.746 + 0.665i)T \)
89 \( 1 + (0.999 - 0.0227i)T \)
97 \( 1 + (0.866 + 0.5i)T \)
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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

−18.50453771101361698364311815996, −17.894600511409961622474516261830, −17.2727176573324594926196116026, −16.63639072495920286646633317115, −16.01564535519028469282645025942, −15.405000533767311072407026075176, −14.71015306015001708652660008989, −13.7578939310323777092385913928, −13.31589941890079222963164681269, −12.755027542921279195425752917905, −11.78128690133527119273575649956, −11.001246210899483066194107966092, −10.4002825658424264659296755004, −9.440597363529441577515930284, −9.07489779261394089756184310113, −7.98903035558097702050258682815, −7.28723055380474084186106874039, −6.39776808309643186732082606165, −6.02580169000763719955680648463, −5.6856530099477347416012047272, −4.812585729013728033936447847512, −3.62056106812396423425687799400, −2.99616198839224082585903506279, −1.65164535201652156470067586698, −0.743143442170474333310241001363, 0.65149992966541622916921156468, 1.326800291036039120890092429521, 2.040693367566973851707993022290, 3.397321258849197300840107926468, 4.00104903100304264942776616088, 4.64724310317378217953755787934, 5.41959612512938304718749876496, 6.12200684083579281154619814082, 6.89714358953036657393726331130, 7.928838733191820592515601360831, 8.97590873818233035154284362151, 9.48917161475420988272049188204, 10.2161069216299512599825902603, 10.51346779839729687013331085029, 11.34674214493496705183561441957, 12.38194315330036163375740732336, 12.54295182301288850434717132158, 13.14421436958165913397202850548, 14.12675081226192317088270537328, 14.56721577923920063902752586603, 15.95429676318031712853854946692, 16.47058776256236709182061131285, 16.988392946593958045036613324541, 17.64932718343280262630746786027, 18.1983730780114952573371617697

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