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Properties

Label 1-4011-4011.3317-r1-0-0
Degree $1$
Conductor $4011$
Sign $0.841 + 0.540i$
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  + (0.677 − 0.735i)2-s + (−0.0825 − 0.996i)4-s + (−0.0825 − 0.996i)5-s + (−0.789 − 0.614i)8-s + (−0.789 − 0.614i)10-s + (0.945 + 0.324i)11-s + (−0.245 − 0.969i)13-s + (−0.986 + 0.164i)16-s + (−0.401 − 0.915i)17-s + (0.789 − 0.614i)19-s + (−0.986 + 0.164i)20-s + (0.879 − 0.475i)22-s + (−0.789 + 0.614i)23-s + (−0.986 + 0.164i)25-s + (−0.879 − 0.475i)26-s + ⋯
L(s)  = 1  + (0.677 − 0.735i)2-s + (−0.0825 − 0.996i)4-s + (−0.0825 − 0.996i)5-s + (−0.789 − 0.614i)8-s + (−0.789 − 0.614i)10-s + (0.945 + 0.324i)11-s + (−0.245 − 0.969i)13-s + (−0.986 + 0.164i)16-s + (−0.401 − 0.915i)17-s + (0.789 − 0.614i)19-s + (−0.986 + 0.164i)20-s + (0.879 − 0.475i)22-s + (−0.789 + 0.614i)23-s + (−0.986 + 0.164i)25-s + (−0.879 − 0.475i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.841 + 0.540i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.841 + 0.540i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4011\)    =    \(3 \cdot 7 \cdot 191\)
Sign: $0.841 + 0.540i$
Analytic conductor: \(431.041\)
Root analytic conductor: \(431.041\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4011} (3317, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4011,\ (1:\ ),\ 0.841 + 0.540i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1802205948 - 0.05288060378i\)
\(L(\frac12)\) \(\approx\) \(-0.1802205948 - 0.05288060378i\)
\(L(1)\) \(\approx\) \(0.8730000813 - 0.8858113231i\)
\(L(1)\) \(\approx\) \(0.8730000813 - 0.8858113231i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
191 \( 1 \)
good2 \( 1 + (0.677 - 0.735i)T \)
5 \( 1 + (-0.0825 - 0.996i)T \)
11 \( 1 + (0.945 + 0.324i)T \)
13 \( 1 + (-0.245 - 0.969i)T \)
17 \( 1 + (-0.401 - 0.915i)T \)
19 \( 1 + (0.789 - 0.614i)T \)
23 \( 1 + (-0.789 + 0.614i)T \)
29 \( 1 + (-0.986 - 0.164i)T \)
31 \( 1 + (-0.879 - 0.475i)T \)
37 \( 1 + (0.879 - 0.475i)T \)
41 \( 1 + (0.677 + 0.735i)T \)
43 \( 1 + (0.546 + 0.837i)T \)
47 \( 1 + (-0.945 - 0.324i)T \)
53 \( 1 + (0.945 + 0.324i)T \)
59 \( 1 + (-0.879 - 0.475i)T \)
61 \( 1 + (-0.401 + 0.915i)T \)
67 \( 1 + (-0.401 + 0.915i)T \)
71 \( 1 + (-0.677 - 0.735i)T \)
73 \( 1 + (0.945 - 0.324i)T \)
79 \( 1 + (-0.986 + 0.164i)T \)
83 \( 1 + (-0.789 - 0.614i)T \)
89 \( 1 + (-0.546 + 0.837i)T \)
97 \( 1 + (0.879 - 0.475i)T \)
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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

−18.69780281307849469031397411208, −18.26358829912541133845743173797, −17.4029494431994158733289661035, −16.74237093463241529774320926651, −16.22920988692391808794731322066, −15.38848903054082657911386187542, −14.70053748020566520784780463288, −14.22995620105414526327522051706, −13.84247056749501644448509122090, −12.81773135216245367959708749041, −12.13028072544275824292138028218, −11.46770795070523044829240921837, −10.90323175475238471251299843040, −9.85588963810143467565938452477, −9.093777120249726788789565967178, −8.36052954344939669964113850352, −7.44665941386877569221688092829, −7.005107095196649188934028880499, −6.12749386411290540377094647139, −5.87281951601258940858513194697, −4.63504888587434064445608990336, −3.854665291263662194388714649093, −3.5076265680981214750404372703, −2.40848467584523413135502873376, −1.645207508269217795853290054141, 0.02331582260702222499307324846, 0.826915515192429409216782047120, 1.55364814338607300661658127112, 2.45883352137073970888595680768, 3.33502625761451428798154911442, 4.17254276400927306328331994117, 4.70148740447329945139878749374, 5.54702155141682205898321364935, 6.00264072755056000705574367417, 7.192219197326481679744177790106, 7.81653622805790830769268003439, 9.03668108971834935095466894523, 9.40872812342791432690217075511, 9.959518503629217892120472697821, 11.116823639696543470118991437410, 11.594367095563764568889140197790, 12.163148241386115993156562331488, 13.03881887510921440584998221347, 13.27109750506004811392155642829, 14.19662777471658708940473277210, 14.85438427631212355857051799124, 15.58323747278503264891025482901, 16.20628884649774462013739745432, 16.987053891902652003566642081219, 17.97965942798250838754369089535

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