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Properties

Label 1-4001-4001.152-r0-0-0
Degree $1$
Conductor $4001$
Sign $0.998 - 0.0469i$
Analytic cond. $18.5805$
Root an. cond. $18.5805$
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  + (−0.809 − 0.587i)2-s + (−0.951 − 0.309i)3-s + (0.309 + 0.951i)4-s + 5-s + (0.587 + 0.809i)6-s + (−0.809 + 0.587i)7-s + (0.309 − 0.951i)8-s + (0.809 + 0.587i)9-s + (−0.809 − 0.587i)10-s − i·11-s − i·12-s + (0.309 + 0.951i)13-s + 14-s + (−0.951 − 0.309i)15-s + (−0.809 + 0.587i)16-s + (0.951 + 0.309i)17-s + ⋯
L(s)  = 1  + (−0.809 − 0.587i)2-s + (−0.951 − 0.309i)3-s + (0.309 + 0.951i)4-s + 5-s + (0.587 + 0.809i)6-s + (−0.809 + 0.587i)7-s + (0.309 − 0.951i)8-s + (0.809 + 0.587i)9-s + (−0.809 − 0.587i)10-s − i·11-s − i·12-s + (0.309 + 0.951i)13-s + 14-s + (−0.951 − 0.309i)15-s + (−0.809 + 0.587i)16-s + (0.951 + 0.309i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0469i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0469i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4001\)
Sign: $0.998 - 0.0469i$
Analytic conductor: \(18.5805\)
Root analytic conductor: \(18.5805\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4001} (152, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4001,\ (0:\ ),\ 0.998 - 0.0469i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.131144179 + 0.02655259240i\)
\(L(\frac12)\) \(\approx\) \(1.131144179 + 0.02655259240i\)
\(L(1)\) \(\approx\) \(0.7197293705 - 0.1134538736i\)
\(L(1)\) \(\approx\) \(0.7197293705 - 0.1134538736i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad4001 \( 1 \)
good2 \( 1 + (-0.809 - 0.587i)T \)
3 \( 1 + (-0.951 - 0.309i)T \)
5 \( 1 + T \)
7 \( 1 + (-0.809 + 0.587i)T \)
11 \( 1 - iT \)
13 \( 1 + (0.309 + 0.951i)T \)
17 \( 1 + (0.951 + 0.309i)T \)
19 \( 1 + (0.309 + 0.951i)T \)
23 \( 1 + (0.951 - 0.309i)T \)
29 \( 1 + (0.809 - 0.587i)T \)
31 \( 1 + (0.809 + 0.587i)T \)
37 \( 1 + (0.587 + 0.809i)T \)
41 \( 1 + (-0.587 - 0.809i)T \)
43 \( 1 + (0.951 - 0.309i)T \)
47 \( 1 + (0.809 + 0.587i)T \)
53 \( 1 + iT \)
59 \( 1 - T \)
61 \( 1 + (0.809 + 0.587i)T \)
67 \( 1 + (-0.951 + 0.309i)T \)
71 \( 1 + (0.309 - 0.951i)T \)
73 \( 1 + iT \)
79 \( 1 + T \)
83 \( 1 + (0.809 + 0.587i)T \)
89 \( 1 + (0.809 + 0.587i)T \)
97 \( 1 + (-0.309 - 0.951i)T \)
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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

−18.1873102354151451967020629876, −17.566829068189981124779273432152, −17.31811195375860205849536936844, −16.54023498179787044764245807366, −16.01128408007264447949387056840, −15.29102268519110566912486257236, −14.67397360343821170281138200373, −13.67278664580451984562756286933, −13.05707752324535718231769622761, −12.34041470911611365411343896500, −11.34868282296361546047474197065, −10.48845306024302127128205558705, −10.24207074829565810687133319883, −9.48750383371815274693466623855, −9.156681119630757967425425993317, −7.80107556219892566108954667913, −7.14090888301722146722795067365, −6.538617258132139827743956273029, −5.94356577937650198645264029577, −5.151002470858048385244165614, −4.69647239121739631373905836913, −3.33450852319057203716597805369, −2.41971257533524741699912272005, −1.14731185021654533638699558780, −0.72908678537868353319131990693, 0.89440451821076553336853703099, 1.38926590091634427538262057116, 2.42751132238401991336841305148, 3.09415019538601367061023321672, 4.080988381318263598156825188019, 5.18966948630264611586651446975, 6.12059699874113302815658824497, 6.29994783413992445804638988836, 7.19328231874224118701648173580, 8.177216112769821286693973018828, 8.9402319644953344705658422668, 9.55398218921875069860189287332, 10.32325707836927757838895329885, 10.6854110107650704213889949540, 11.67506307214699315765735392248, 12.18507917639522953666254905884, 12.722945572644001873221588941911, 13.59504965475199844436751464068, 14.01034155541545360300649660091, 15.417602144459408729612230473539, 16.18007171049426821000900785119, 16.74040299685550271681252056516, 17.00303361111459416985585545226, 17.91868411636405778351758973084, 18.58951138822999093585620353703

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