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Properties

Label 1-4011-4011.500-r0-0-0
Degree $1$
Conductor $4011$
Sign $-0.373 - 0.927i$
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  + (0.949 − 0.314i)2-s + (0.802 − 0.596i)4-s + (0.975 − 0.218i)5-s + (0.574 − 0.818i)8-s + (0.857 − 0.514i)10-s + (−0.635 − 0.771i)11-s + (0.277 − 0.960i)13-s + (0.287 − 0.957i)16-s + (−0.889 + 0.456i)17-s + (0.857 + 0.514i)19-s + (0.652 − 0.757i)20-s + (−0.846 − 0.533i)22-s + (−0.391 − 0.920i)23-s + (0.904 − 0.426i)25-s + (−0.0385 − 0.999i)26-s + ⋯
L(s)  = 1  + (0.949 − 0.314i)2-s + (0.802 − 0.596i)4-s + (0.975 − 0.218i)5-s + (0.574 − 0.818i)8-s + (0.857 − 0.514i)10-s + (−0.635 − 0.771i)11-s + (0.277 − 0.960i)13-s + (0.287 − 0.957i)16-s + (−0.889 + 0.456i)17-s + (0.857 + 0.514i)19-s + (0.652 − 0.757i)20-s + (−0.846 − 0.533i)22-s + (−0.391 − 0.920i)23-s + (0.904 − 0.426i)25-s + (−0.0385 − 0.999i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.373 - 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.373 - 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4011\)    =    \(3 \cdot 7 \cdot 191\)
Sign: $-0.373 - 0.927i$
Analytic conductor: \(18.6270\)
Root analytic conductor: \(18.6270\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4011} (500, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4011,\ (0:\ ),\ -0.373 - 0.927i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.161122482 - 3.201302500i\)
\(L(\frac12)\) \(\approx\) \(2.161122482 - 3.201302500i\)
\(L(1)\) \(\approx\) \(1.928340646 - 0.9735745511i\)
\(L(1)\) \(\approx\) \(1.928340646 - 0.9735745511i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
191 \( 1 \)
good2 \( 1 + (0.949 - 0.314i)T \)
5 \( 1 + (0.975 - 0.218i)T \)
11 \( 1 + (-0.635 - 0.771i)T \)
13 \( 1 + (0.277 - 0.960i)T \)
17 \( 1 + (-0.889 + 0.456i)T \)
19 \( 1 + (0.857 + 0.514i)T \)
23 \( 1 + (-0.391 - 0.920i)T \)
29 \( 1 + (0.518 - 0.854i)T \)
31 \( 1 + (-0.716 + 0.697i)T \)
37 \( 1 + (0.716 + 0.697i)T \)
41 \( 1 + (-0.401 - 0.915i)T \)
43 \( 1 + (0.991 + 0.131i)T \)
47 \( 1 + (-0.834 + 0.551i)T \)
53 \( 1 + (-0.930 + 0.366i)T \)
59 \( 1 + (-0.441 - 0.897i)T \)
61 \( 1 + (0.709 - 0.705i)T \)
67 \( 1 + (0.775 + 0.631i)T \)
71 \( 1 + (0.213 - 0.976i)T \)
73 \( 1 + (-0.266 + 0.963i)T \)
79 \( 1 + (0.999 - 0.0220i)T \)
83 \( 1 + (0.601 + 0.799i)T \)
89 \( 1 + (-0.997 - 0.0770i)T \)
97 \( 1 + (-0.371 - 0.928i)T \)
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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

−18.40814974177188036876619749594, −17.90703299252498836161540139604, −17.38185575251281691879920582654, −16.37710741296953593757004587595, −16.01341359030168781762333089961, −15.163361072616115374767897863099, −14.52604769751708417935905509528, −13.84118119927940889038402829243, −13.33260681179493738943890120805, −12.82423671024265706970952118842, −11.87298564887294961434715271434, −11.24489219269964372861307875151, −10.5979318088191462429867371012, −9.576324865889114217135697477975, −9.152242556050822154714324055335, −8.000513446425764898849788064910, −7.186686625901522187711118289089, −6.734222778770497001948410994278, −5.93237765638233569695708407493, −5.18846918468235600507455850381, −4.65387782403765834653131844624, −3.74430979103599792694134313723, −2.78167335380490928555640311785, −2.18264173410363448642085870529, −1.449972521150231305062938651501, 0.68520364640613192787263505036, 1.602816441917575445120503562837, 2.4827305178813640848982306793, 3.05448261794649981463868709183, 3.94720716589272617126296204082, 4.9037651275247862252217962268, 5.46262896581730665445049075107, 6.13107104775040588405335354218, 6.62234584292651022885020867253, 7.819885009095588545636353655984, 8.432217117265649988933091288890, 9.48705961048442837520618848951, 10.16089299837771467028274503557, 10.77127742633782028881004394457, 11.29703917557841512067402147183, 12.47933225649663583750345510281, 12.74782721410099078578790594577, 13.55147988717808925144053700935, 13.98002715967222964482692427530, 14.647437785443386545698460365033, 15.61893536188712759803593380008, 15.98958296855498177139864283621, 16.799274619061507171858951136808, 17.64846802031845199513433883235, 18.29851267896484203036158543555

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