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Properties

Label 1-4011-4011.47-r1-0-0
Degree $1$
Conductor $4011$
Sign $0.202 - 0.979i$
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  + (−0.583 − 0.812i)2-s + (−0.319 + 0.947i)4-s + (0.975 + 0.218i)5-s + (0.956 − 0.293i)8-s + (−0.391 − 0.920i)10-s + (0.635 − 0.771i)11-s + (0.999 + 0.0330i)13-s + (−0.795 − 0.605i)16-s + (0.159 − 0.987i)17-s + (0.391 − 0.920i)19-s + (−0.518 + 0.854i)20-s + (−0.997 − 0.0660i)22-s + (0.857 − 0.514i)23-s + (0.904 + 0.426i)25-s + (−0.556 − 0.831i)26-s + ⋯
L(s)  = 1  + (−0.583 − 0.812i)2-s + (−0.319 + 0.947i)4-s + (0.975 + 0.218i)5-s + (0.956 − 0.293i)8-s + (−0.391 − 0.920i)10-s + (0.635 − 0.771i)11-s + (0.999 + 0.0330i)13-s + (−0.795 − 0.605i)16-s + (0.159 − 0.987i)17-s + (0.391 − 0.920i)19-s + (−0.518 + 0.854i)20-s + (−0.997 − 0.0660i)22-s + (0.857 − 0.514i)23-s + (0.904 + 0.426i)25-s + (−0.556 − 0.831i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.202 - 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.202 - 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4011\)    =    \(3 \cdot 7 \cdot 191\)
Sign: $0.202 - 0.979i$
Analytic conductor: \(431.041\)
Root analytic conductor: \(431.041\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4011} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4011,\ (1:\ ),\ 0.202 - 0.979i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.407392893 - 1.959912734i\)
\(L(\frac12)\) \(\approx\) \(2.407392893 - 1.959912734i\)
\(L(1)\) \(\approx\) \(1.108351270 - 0.4812558534i\)
\(L(1)\) \(\approx\) \(1.108351270 - 0.4812558534i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
191 \( 1 \)
good2 \( 1 + (-0.583 - 0.812i)T \)
5 \( 1 + (0.975 + 0.218i)T \)
11 \( 1 + (0.635 - 0.771i)T \)
13 \( 1 + (0.999 + 0.0330i)T \)
17 \( 1 + (0.159 - 0.987i)T \)
19 \( 1 + (0.391 - 0.920i)T \)
23 \( 1 + (0.857 - 0.514i)T \)
29 \( 1 + (0.652 - 0.757i)T \)
31 \( 1 + (0.716 + 0.697i)T \)
37 \( 1 + (-0.716 + 0.697i)T \)
41 \( 1 + (0.401 - 0.915i)T \)
43 \( 1 + (0.431 + 0.901i)T \)
47 \( 1 + (-0.266 + 0.963i)T \)
53 \( 1 + (-0.537 - 0.843i)T \)
59 \( 1 + (0.884 - 0.466i)T \)
61 \( 1 + (0.988 + 0.153i)T \)
67 \( 1 + (0.840 + 0.542i)T \)
71 \( 1 + (0.863 - 0.504i)T \)
73 \( 1 + (-0.834 + 0.551i)T \)
79 \( 1 + (0.329 - 0.944i)T \)
83 \( 1 + (0.0165 - 0.999i)T \)
89 \( 1 + (0.381 + 0.924i)T \)
97 \( 1 + (0.768 + 0.639i)T \)
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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

−18.244077030286205879026388820495, −17.68916212791154477737278749111, −17.08982904587624517967201586189, −16.63150805242461287790309293129, −15.7858079167650095909300858964, −15.13450000613856882888999806858, −14.36930302605423648990515575367, −13.93402075777727016244316834823, −13.10205481911756761174678821001, −12.48992110923750449136596910218, −11.41966713015605214517863004486, −10.55066384314058232321640487117, −10.03692825775262040795225078708, −9.36588461169584326850223875880, −8.712301017637839227793057292799, −8.12076549420635513245774283435, −7.12806601688972092499911413154, −6.51941585763469655651451963755, −5.84625154723656169195553243174, −5.288089413157151765540295803458, −4.36231494173301886479832207741, −3.51470406307911681760883400318, −2.14757414038490899175262793127, −1.44814482820469172584549493888, −0.88477351509080151011493111449, 0.787790501722471488350454185669, 1.03934287132607484348872547944, 2.18553733227642395676174932721, 2.92769789416047493869606144285, 3.464622255649307132879174393281, 4.587334822451244706474903362582, 5.280846253216272147775830340188, 6.42803757149840045681726841389, 6.79235204591070467007606719695, 7.87863172125108029321860237460, 8.80113212069640430968223186569, 9.06958803036888576598587136656, 9.89660189192042509121876732824, 10.53605807130188639488391539606, 11.32956692429767048378261132743, 11.639307922068677986597594821009, 12.7414702549394668952128947261, 13.339252305009246210031646905662, 13.92896156542108867852958110367, 14.40286034960467356484570954551, 15.81359466495910537774881431161, 16.15190802713990639502790222731, 17.13150693345555252180383162722, 17.58041522065806413143969102862, 18.125547607242976058420761185005

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