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Properties

Label 1-4011-4011.1094-r0-0-0
Degree $1$
Conductor $4011$
Sign $0.594 - 0.804i$
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  + (−0.993 + 0.110i)2-s + (0.975 − 0.218i)4-s + (0.298 − 0.954i)5-s + (−0.945 + 0.324i)8-s + (−0.191 + 0.981i)10-s + (0.350 − 0.936i)11-s + (0.789 − 0.614i)13-s + (0.904 − 0.426i)16-s + (0.998 − 0.0550i)17-s + (0.191 + 0.981i)19-s + (0.0825 − 0.996i)20-s + (−0.245 + 0.969i)22-s + (0.191 + 0.981i)23-s + (−0.821 − 0.569i)25-s + (−0.716 + 0.697i)26-s + ⋯
L(s)  = 1  + (−0.993 + 0.110i)2-s + (0.975 − 0.218i)4-s + (0.298 − 0.954i)5-s + (−0.945 + 0.324i)8-s + (−0.191 + 0.981i)10-s + (0.350 − 0.936i)11-s + (0.789 − 0.614i)13-s + (0.904 − 0.426i)16-s + (0.998 − 0.0550i)17-s + (0.191 + 0.981i)19-s + (0.0825 − 0.996i)20-s + (−0.245 + 0.969i)22-s + (0.191 + 0.981i)23-s + (−0.821 − 0.569i)25-s + (−0.716 + 0.697i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.594 - 0.804i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.594 - 0.804i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4011\)    =    \(3 \cdot 7 \cdot 191\)
Sign: $0.594 - 0.804i$
Analytic conductor: \(18.6270\)
Root analytic conductor: \(18.6270\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4011} (1094, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4011,\ (0:\ ),\ 0.594 - 0.804i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.303303063 - 0.6572965688i\)
\(L(\frac12)\) \(\approx\) \(1.303303063 - 0.6572965688i\)
\(L(1)\) \(\approx\) \(0.8660968476 - 0.1918210974i\)
\(L(1)\) \(\approx\) \(0.8660968476 - 0.1918210974i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
191 \( 1 \)
good2 \( 1 + (-0.993 + 0.110i)T \)
5 \( 1 + (0.298 - 0.954i)T \)
11 \( 1 + (0.350 - 0.936i)T \)
13 \( 1 + (0.789 - 0.614i)T \)
17 \( 1 + (0.998 - 0.0550i)T \)
19 \( 1 + (0.191 + 0.981i)T \)
23 \( 1 + (0.191 + 0.981i)T \)
29 \( 1 + (-0.0825 - 0.996i)T \)
31 \( 1 + (0.962 + 0.272i)T \)
37 \( 1 + (0.962 - 0.272i)T \)
41 \( 1 + (-0.401 - 0.915i)T \)
43 \( 1 + (-0.879 + 0.475i)T \)
47 \( 1 + (0.635 + 0.771i)T \)
53 \( 1 + (0.350 - 0.936i)T \)
59 \( 1 + (0.962 + 0.272i)T \)
61 \( 1 + (0.998 + 0.0550i)T \)
67 \( 1 + (0.451 + 0.892i)T \)
71 \( 1 + (-0.401 - 0.915i)T \)
73 \( 1 + (-0.635 + 0.771i)T \)
79 \( 1 + (0.904 - 0.426i)T \)
83 \( 1 + (0.945 - 0.324i)T \)
89 \( 1 + (0.0275 + 0.999i)T \)
97 \( 1 + (0.245 - 0.969i)T \)
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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

−18.559973354666818091795429860898, −18.03569408436480152094208783861, −17.32639097972899477603381065243, −16.69334466421416483887592646177, −16.01829822957948674259750027983, −15.07255511129726077559115544750, −14.81047416407631132741979615406, −13.891415278383312150278367841969, −13.06914836211212782457633236117, −12.1197461728751538571636228054, −11.56955152411617144226745405577, −10.881616567032217147316658069322, −10.190722263534965153286374792411, −9.68447339471185381758741281752, −8.93988177896981563360100501867, −8.1834556932958188730282329821, −7.31962126271211360916816455877, −6.73034229860124614784428735979, −6.31426590671660419543493589122, −5.256631202181815825668201419440, −4.141310531521299234200238938546, −3.24583950980195151307448954329, −2.55215844484862543615019304126, −1.7707715642399055866871010220, −0.905757318438710941837409618618, 0.82397076847258177831575486139, 1.14865771821260405074535294121, 2.1619329902095968417251997657, 3.2510855416593137045028686276, 3.87804970063056134537594688728, 5.243734801548620467947659475523, 5.80130953018913519606247308379, 6.27325672620956552407847391645, 7.445819540948544443497599286016, 8.1989272121672522285891962721, 8.4611631421518610336148231225, 9.395689185694920502222586513166, 9.91903927656123589859250088583, 10.59787406015611107404210709330, 11.618582423105319615651452593867, 11.88462817795379048937046337092, 12.88936598792812306348041161700, 13.57434330397082127002290708219, 14.33413093110343549277580581218, 15.1955232551841311654695100972, 16.075829765445910415847861550239, 16.29519447007639275837464066024, 17.09496461231642662340193932677, 17.578738119980854592078371085051, 18.347090993124065109124142513131

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