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Properties

Label 1-4100-4100.863-r0-0-0
Degree $1$
Conductor $4100$
Sign $0.436 + 0.899i$
Analytic cond. $19.0403$
Root an. cond. $19.0403$
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)3-s + (−0.809 − 0.587i)7-s + (0.309 − 0.951i)9-s + (−0.587 − 0.809i)11-s + (0.309 + 0.951i)13-s + (0.309 − 0.951i)17-s + (−0.951 + 0.309i)19-s − 21-s + (−0.587 + 0.809i)23-s + (−0.309 − 0.951i)27-s + (−0.951 + 0.309i)29-s + (−0.309 + 0.951i)31-s + (−0.951 − 0.309i)33-s + (−0.587 + 0.809i)37-s + (0.809 + 0.587i)39-s + ⋯
L(s)  = 1  + (0.809 − 0.587i)3-s + (−0.809 − 0.587i)7-s + (0.309 − 0.951i)9-s + (−0.587 − 0.809i)11-s + (0.309 + 0.951i)13-s + (0.309 − 0.951i)17-s + (−0.951 + 0.309i)19-s − 21-s + (−0.587 + 0.809i)23-s + (−0.309 − 0.951i)27-s + (−0.951 + 0.309i)29-s + (−0.309 + 0.951i)31-s + (−0.951 − 0.309i)33-s + (−0.587 + 0.809i)37-s + (0.809 + 0.587i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4100 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.436 + 0.899i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4100 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.436 + 0.899i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4100\)    =    \(2^{2} \cdot 5^{2} \cdot 41\)
Sign: $0.436 + 0.899i$
Analytic conductor: \(19.0403\)
Root analytic conductor: \(19.0403\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4100} (863, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4100,\ (0:\ ),\ 0.436 + 0.899i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7429761304 + 0.4652118043i\)
\(L(\frac12)\) \(\approx\) \(0.7429761304 + 0.4652118043i\)
\(L(1)\) \(\approx\) \(1.009314766 - 0.2113219095i\)
\(L(1)\) \(\approx\) \(1.009314766 - 0.2113219095i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
41 \( 1 \)
good3 \( 1 + (0.809 - 0.587i)T \)
7 \( 1 + (-0.809 - 0.587i)T \)
11 \( 1 + (-0.587 - 0.809i)T \)
13 \( 1 + (0.309 + 0.951i)T \)
17 \( 1 + (0.309 - 0.951i)T \)
19 \( 1 + (-0.951 + 0.309i)T \)
23 \( 1 + (-0.587 + 0.809i)T \)
29 \( 1 + (-0.951 + 0.309i)T \)
31 \( 1 + (-0.309 + 0.951i)T \)
37 \( 1 + (-0.587 + 0.809i)T \)
43 \( 1 + (-0.587 - 0.809i)T \)
47 \( 1 + (-0.309 + 0.951i)T \)
53 \( 1 + (0.309 + 0.951i)T \)
59 \( 1 + (0.309 + 0.951i)T \)
61 \( 1 + (-0.309 + 0.951i)T \)
67 \( 1 + (0.809 - 0.587i)T \)
71 \( 1 + (0.587 - 0.809i)T \)
73 \( 1 + (0.951 - 0.309i)T \)
79 \( 1 + (0.587 + 0.809i)T \)
83 \( 1 + (-0.587 + 0.809i)T \)
89 \( 1 + (0.951 + 0.309i)T \)
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

−18.60111096397988678509347622999, −17.61592498439747649377275115574, −16.854723521607416436825778882756, −16.087635050047260425788498335481, −15.50861256053927744369749295507, −14.95646928583852020058699040574, −14.55026642145538484293930163144, −13.35540115559636112881019125609, −12.8804294146659479789287727411, −12.52679281713482364662729344403, −11.2935829208874380895755025135, −10.49586229366942316479790181633, −9.99445512378244161373166992833, −9.431023453569082881167329471616, −8.45935574630377602200796491504, −8.154394112467823011912762505743, −7.239201490355258602493010752911, −6.30495930091431385113594124506, −5.56171379485242702170589846215, −4.79818960646084571396919994868, −3.82566424202257713846510474187, −3.38883370268651202270843538009, −2.25662200941656922152700007089, −2.053949270303240801265371436298, −0.219421122216253382639803746335, 1.00526622958060301784469933823, 1.863449343380860335638202424, 2.77401058590661864272411992542, 3.508568706525038213871610152852, 3.99192793942802503308473827224, 5.1650847105701987699846182282, 6.12496152783731791355480030218, 6.73267258644949000974198298356, 7.39112153251004716784923862760, 8.07416762145251570407702916187, 8.90936339157459084297198675678, 9.39200489776815633396497852126, 10.22236253366753613389544991982, 10.95530840144077000947720583312, 11.86933175442884543188968339247, 12.50986798935478464452890564875, 13.31926339820583028804580353374, 13.76723210840059188457540596147, 14.180601149817045270812691359075, 15.170186026727191565133732252833, 15.85914655330498886564523151384, 16.50532866788377237555003613098, 17.08761612108943233509306584100, 18.23267003380336326374988885143, 18.56576901212193078235433535596

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