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Properties

Label 1-4011-4011.3236-r1-0-0
Degree $1$
Conductor $4011$
Sign $0.789 + 0.613i$
Analytic cond. $431.041$
Root an. cond. $431.041$
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  + (0.592 − 0.805i)2-s + (−0.298 − 0.954i)4-s + (−0.975 − 0.218i)5-s + (−0.945 − 0.324i)8-s + (−0.754 + 0.656i)10-s + (−0.635 + 0.771i)11-s + (0.789 + 0.614i)13-s + (−0.821 + 0.569i)16-s + (−0.451 − 0.892i)17-s + (−0.754 − 0.656i)19-s + (0.0825 + 0.996i)20-s + (0.245 + 0.969i)22-s + (0.754 + 0.656i)23-s + (0.904 + 0.426i)25-s + (0.962 − 0.272i)26-s + ⋯
L(s)  = 1  + (0.592 − 0.805i)2-s + (−0.298 − 0.954i)4-s + (−0.975 − 0.218i)5-s + (−0.945 − 0.324i)8-s + (−0.754 + 0.656i)10-s + (−0.635 + 0.771i)11-s + (0.789 + 0.614i)13-s + (−0.821 + 0.569i)16-s + (−0.451 − 0.892i)17-s + (−0.754 − 0.656i)19-s + (0.0825 + 0.996i)20-s + (0.245 + 0.969i)22-s + (0.754 + 0.656i)23-s + (0.904 + 0.426i)25-s + (0.962 − 0.272i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.789 + 0.613i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.789 + 0.613i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4011\)    =    \(3 \cdot 7 \cdot 191\)
Sign: $0.789 + 0.613i$
Analytic conductor: \(431.041\)
Root analytic conductor: \(431.041\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4011} (3236, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4011,\ (1:\ ),\ 0.789 + 0.613i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3392781474 + 0.1162466224i\)
\(L(\frac12)\) \(\approx\) \(0.3392781474 + 0.1162466224i\)
\(L(1)\) \(\approx\) \(0.8135464699 - 0.5222435459i\)
\(L(1)\) \(\approx\) \(0.8135464699 - 0.5222435459i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
191 \( 1 \)
good2 \( 1 + (0.592 - 0.805i)T \)
5 \( 1 + (-0.975 - 0.218i)T \)
11 \( 1 + (-0.635 + 0.771i)T \)
13 \( 1 + (0.789 + 0.614i)T \)
17 \( 1 + (-0.451 - 0.892i)T \)
19 \( 1 + (-0.754 - 0.656i)T \)
23 \( 1 + (0.754 + 0.656i)T \)
29 \( 1 + (0.0825 - 0.996i)T \)
31 \( 1 + (0.716 + 0.697i)T \)
37 \( 1 + (0.716 - 0.697i)T \)
41 \( 1 + (0.401 - 0.915i)T \)
43 \( 1 + (-0.879 - 0.475i)T \)
47 \( 1 + (-0.350 - 0.936i)T \)
53 \( 1 + (-0.635 + 0.771i)T \)
59 \( 1 + (-0.716 - 0.697i)T \)
61 \( 1 + (0.451 - 0.892i)T \)
67 \( 1 + (-0.998 + 0.0550i)T \)
71 \( 1 + (0.401 - 0.915i)T \)
73 \( 1 + (0.350 - 0.936i)T \)
79 \( 1 + (-0.821 + 0.569i)T \)
83 \( 1 + (-0.945 - 0.324i)T \)
89 \( 1 + (-0.851 - 0.523i)T \)
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

−18.29588940212806087161055248569, −17.42051501461407250846771308022, −16.54106468523931359932291520106, −16.22581032605840238315455437542, −15.395181604414454535565347448340, −14.95957112823851075451181899755, −14.378976949232459162751302947162, −13.31790917369832661467778318106, −12.94782006573912559835041384570, −12.31338741297882021689621409059, −11.23546785537949377404548889553, −10.98709923205899947192359006067, −9.992396697318733680786618892943, −8.66398048606141928959757875519, −8.32476752504646180942308616231, −7.88647011142365799306078895243, −6.87192527005105805595210534934, −6.25663858459105241816055901009, −5.63393013095112968300375286086, −4.58165965342615937747535128502, −4.1281974734662867436261017508, −3.15003923134905861301064116080, −2.808081428188048233738132361382, −1.20708902320494739563844760529, −0.06379497850495852923504094181, 0.64716534382275527798653694856, 1.69795629185239840341236516169, 2.533084067229519717321990407118, 3.309378333113960791950080802068, 4.16594525159905141772131060749, 4.673063062958370180768016630892, 5.29825926681569820490741523800, 6.43165781947084741399594572635, 7.041235746489672678355639362860, 7.94001953687863157665654906497, 8.87890865800890355559560538083, 9.33337637651540684429039556555, 10.32108616770806537435897645192, 11.035193031629717172669708771517, 11.50259060189919220090880065628, 12.16574543442491164365007120263, 12.88443479739549062786441544142, 13.44902723526321652901875048897, 14.12304862540463420884542478047, 15.130002871547878960911170849479, 15.49477470529743489106641673984, 16.014139930877331495305896409265, 17.09382186948929955235426163341, 17.90358422601968283304649209289, 18.62873462826563603001109957775

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