Properties

Label 1-4011-4011.311-r0-0-0
Degree $1$
Conductor $4011$
Sign $0.637 - 0.770i$
Analytic cond. $18.6270$
Root an. cond. $18.6270$
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  + (0.556 − 0.831i)2-s + (−0.381 − 0.924i)4-s + (0.0275 − 0.999i)5-s + (−0.980 − 0.197i)8-s + (−0.815 − 0.578i)10-s + (−0.993 + 0.110i)11-s + (0.518 + 0.854i)13-s + (−0.709 + 0.705i)16-s + (−0.693 + 0.720i)17-s + (−0.815 + 0.578i)19-s + (−0.934 + 0.355i)20-s + (−0.461 + 0.887i)22-s + (0.999 + 0.0110i)23-s + (−0.998 − 0.0550i)25-s + (0.999 + 0.0440i)26-s + ⋯
L(s)  = 1  + (0.556 − 0.831i)2-s + (−0.381 − 0.924i)4-s + (0.0275 − 0.999i)5-s + (−0.980 − 0.197i)8-s + (−0.815 − 0.578i)10-s + (−0.993 + 0.110i)11-s + (0.518 + 0.854i)13-s + (−0.709 + 0.705i)16-s + (−0.693 + 0.720i)17-s + (−0.815 + 0.578i)19-s + (−0.934 + 0.355i)20-s + (−0.461 + 0.887i)22-s + (0.999 + 0.0110i)23-s + (−0.998 − 0.0550i)25-s + (0.999 + 0.0440i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.637 - 0.770i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.637 - 0.770i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4011\)    =    \(3 \cdot 7 \cdot 191\)
Sign: $0.637 - 0.770i$
Analytic conductor: \(18.6270\)
Root analytic conductor: \(18.6270\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4011} (311, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4011,\ (0:\ ),\ 0.637 - 0.770i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.377239056 - 0.6475482730i\)
\(L(\frac12)\) \(\approx\) \(1.377239056 - 0.6475482730i\)
\(L(1)\) \(\approx\) \(0.9790768308 - 0.6003565692i\)
\(L(1)\) \(\approx\) \(0.9790768308 - 0.6003565692i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
191 \( 1 \)
good2 \( 1 + (0.556 - 0.831i)T \)
5 \( 1 + (0.0275 - 0.999i)T \)
11 \( 1 + (-0.993 + 0.110i)T \)
13 \( 1 + (0.518 + 0.854i)T \)
17 \( 1 + (-0.693 + 0.720i)T \)
19 \( 1 + (-0.815 + 0.578i)T \)
23 \( 1 + (0.999 + 0.0110i)T \)
29 \( 1 + (-0.0495 + 0.998i)T \)
31 \( 1 + (-0.635 - 0.771i)T \)
37 \( 1 + (0.635 - 0.771i)T \)
41 \( 1 + (0.245 - 0.969i)T \)
43 \( 1 + (-0.956 - 0.293i)T \)
47 \( 1 + (0.00551 + 0.999i)T \)
53 \( 1 + (-0.202 + 0.979i)T \)
59 \( 1 + (0.930 - 0.366i)T \)
61 \( 1 + (0.899 - 0.436i)T \)
67 \( 1 + (0.528 + 0.849i)T \)
71 \( 1 + (0.768 - 0.639i)T \)
73 \( 1 + (-0.952 + 0.303i)T \)
79 \( 1 + (-0.889 - 0.456i)T \)
83 \( 1 + (0.490 - 0.871i)T \)
89 \( 1 + (0.996 - 0.0880i)T \)
97 \( 1 + (-0.894 + 0.446i)T \)
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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

−18.24063863676181319192539859257, −17.97930781441387896593829535451, −17.22969877515857684954309456934, −16.35720821871885547328183057458, −15.66419184726861555481200948847, −15.13675530754656763974082062208, −14.7398074662138750708882859479, −13.73143401582037058491132171294, −13.20441727080868465706850034985, −12.85025536096278006889731693209, −11.56213230302785263090726901842, −11.17652496442111500936456164552, −10.32876410786451694694487864809, −9.54467177607908321010087292741, −8.48062024845189423827746450653, −8.08613855336078298732348842613, −7.105846875900795279704409528173, −6.741446402354689917364064957144, −5.90387383711784577942558379151, −5.19052803635904352875885927539, −4.48653081247667129439237976457, −3.45529496217951693096836597779, −2.888236879976165432379840920224, −2.23310460258814498764482009665, −0.45145394317823381102535235838, 0.76712837168616011008189054767, 1.80160830662116894146149438258, 2.21746567770191939194832494989, 3.41913269004108771637892427198, 4.15140074272823083979270273925, 4.70926537498367937483144878049, 5.510418280861911610625244645, 6.09739674542038888654794010523, 7.0649179932232763647795146281, 8.167640787739762945951688667119, 8.86399287593476499864306175266, 9.31027350320254014922136739525, 10.29762793910703958372888798051, 10.88827488213333929560586459750, 11.49778619263074867200159439848, 12.40712297839017486816887528958, 13.05924629939255346140465253578, 13.13485004530455348252334257525, 14.208240642121931262211032435, 14.835928955493955844483269733731, 15.65925433389815394097776965362, 16.22232225489537348069125586817, 17.087448450802295700654773197332, 17.75463194807674188636327956150, 18.71079943357415802252739002226

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