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Properties

Label 1-4100-4100.111-r0-0-0
Degree $1$
Conductor $4100$
Sign $-0.498 + 0.866i$
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  + (−0.156 − 0.987i)3-s + (−0.453 + 0.891i)7-s + (−0.951 + 0.309i)9-s + (0.453 − 0.891i)11-s + (−0.707 − 0.707i)13-s + (0.453 + 0.891i)17-s + (−0.987 + 0.156i)19-s + (0.951 + 0.309i)21-s + (−0.809 − 0.587i)23-s + (0.453 + 0.891i)27-s + (0.453 − 0.891i)29-s + (0.309 − 0.951i)31-s + (−0.951 − 0.309i)33-s + (0.309 − 0.951i)37-s + (−0.587 + 0.809i)39-s + ⋯
L(s)  = 1  + (−0.156 − 0.987i)3-s + (−0.453 + 0.891i)7-s + (−0.951 + 0.309i)9-s + (0.453 − 0.891i)11-s + (−0.707 − 0.707i)13-s + (0.453 + 0.891i)17-s + (−0.987 + 0.156i)19-s + (0.951 + 0.309i)21-s + (−0.809 − 0.587i)23-s + (0.453 + 0.891i)27-s + (0.453 − 0.891i)29-s + (0.309 − 0.951i)31-s + (−0.951 − 0.309i)33-s + (0.309 − 0.951i)37-s + (−0.587 + 0.809i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4100 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.498 + 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4100 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.498 + 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4100\)    =    \(2^{2} \cdot 5^{2} \cdot 41\)
Sign: $-0.498 + 0.866i$
Analytic conductor: \(19.0403\)
Root analytic conductor: \(19.0403\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4100} (111, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4100,\ (0:\ ),\ -0.498 + 0.866i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.007002958145 + 0.01210260177i\)
\(L(\frac12)\) \(\approx\) \(0.007002958145 + 0.01210260177i\)
\(L(1)\) \(\approx\) \(0.7267161053 - 0.2358545361i\)
\(L(1)\) \(\approx\) \(0.7267161053 - 0.2358545361i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
41 \( 1 \)
good3 \( 1 + (-0.156 - 0.987i)T \)
7 \( 1 + (-0.453 + 0.891i)T \)
11 \( 1 + (0.453 - 0.891i)T \)
13 \( 1 + (-0.707 - 0.707i)T \)
17 \( 1 + (0.453 + 0.891i)T \)
19 \( 1 + (-0.987 + 0.156i)T \)
23 \( 1 + (-0.809 - 0.587i)T \)
29 \( 1 + (0.453 - 0.891i)T \)
31 \( 1 + (0.309 - 0.951i)T \)
37 \( 1 + (0.309 - 0.951i)T \)
43 \( 1 + (0.951 + 0.309i)T \)
47 \( 1 + (-0.156 + 0.987i)T \)
53 \( 1 + (-0.453 + 0.891i)T \)
59 \( 1 - T \)
61 \( 1 - iT \)
67 \( 1 + (0.707 - 0.707i)T \)
71 \( 1 + (0.707 + 0.707i)T \)
73 \( 1 + (-0.951 - 0.309i)T \)
79 \( 1 + (0.156 + 0.987i)T \)
83 \( 1 + (0.809 - 0.587i)T \)
89 \( 1 + (-0.707 - 0.707i)T \)
97 \( 1 + (0.891 + 0.453i)T \)
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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

−19.12366809588093453677156163829, −17.904714432220631735972769736030, −17.4557636213859965797312476621, −16.676465376918347209616151931834, −16.35064506499966433755777372748, −15.54644230294053005282059504254, −14.81601161352938953420570243088, −14.19431888155825506196450076566, −13.694512642922392455061563006448, −12.56806360874369941020607320054, −12.00889451281528733459296945843, −11.31034577863600947260007607752, −10.40905790349062925308457706200, −9.97621644820639145491705213171, −9.40916091588663689375361813005, −8.69803624405507988066178126038, −7.66898296439940656255051803764, −6.89128517344013858209996747404, −6.39404030405547967579585191105, −5.24043614937770038287053433082, −4.62225042834412794754284409258, −4.06409115883695421247062869970, −3.29316295746920277802964241998, −2.414247429505795942623237220157, −1.31428370215858359622416712181, 0.00446954836639921705532876785, 0.984778108784008644061674124382, 2.14001506243587820178225220666, 2.59271444474094427143558051402, 3.50177249651320092811935883538, 4.485522535460969493073032411212, 5.632608953474545214295240179701, 6.11772780917944667263810538010, 6.43374641483330436348062979421, 7.74419356193142911278323238084, 8.05873812773366371402455107662, 8.82818088751321263947552101006, 9.60040219320398235755277458975, 10.51607348797011834467333919349, 11.22811620366706757487462479265, 12.00877432078372437501691914362, 12.63949429229322747882950767956, 12.87133575920180456290528370721, 13.96673284621849923167479451250, 14.44941385738300537373690499321, 15.22835299721975557196949408908, 15.96514243217569501216074241404, 16.92261790298132740344407179967, 17.20382369618153607866097781764, 18.07794467729067216547974634646

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