Properties

Label 1-4011-4011.20-r0-0-0
Degree $1$
Conductor $4011$
Sign $0.604 + 0.796i$
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  + (−0.965 + 0.261i)2-s + (0.863 − 0.504i)4-s + (−0.401 + 0.915i)5-s + (−0.701 + 0.712i)8-s + (0.148 − 0.988i)10-s + (0.0825 + 0.996i)11-s + (0.574 − 0.818i)13-s + (0.490 − 0.871i)16-s + (0.431 + 0.901i)17-s + (0.148 + 0.988i)19-s + (0.115 + 0.993i)20-s + (−0.340 − 0.940i)22-s + (0.461 − 0.887i)23-s + (−0.677 − 0.735i)25-s + (−0.340 + 0.940i)26-s + ⋯
L(s)  = 1  + (−0.965 + 0.261i)2-s + (0.863 − 0.504i)4-s + (−0.401 + 0.915i)5-s + (−0.701 + 0.712i)8-s + (0.148 − 0.988i)10-s + (0.0825 + 0.996i)11-s + (0.574 − 0.818i)13-s + (0.490 − 0.871i)16-s + (0.431 + 0.901i)17-s + (0.148 + 0.988i)19-s + (0.115 + 0.993i)20-s + (−0.340 − 0.940i)22-s + (0.461 − 0.887i)23-s + (−0.677 − 0.735i)25-s + (−0.340 + 0.940i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.604 + 0.796i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.604 + 0.796i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4011\)    =    \(3 \cdot 7 \cdot 191\)
Sign: $0.604 + 0.796i$
Analytic conductor: \(18.6270\)
Root analytic conductor: \(18.6270\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4011} (20, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4011,\ (0:\ ),\ 0.604 + 0.796i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9415016011 + 0.4670798515i\)
\(L(\frac12)\) \(\approx\) \(0.9415016011 + 0.4670798515i\)
\(L(1)\) \(\approx\) \(0.6913937367 + 0.2058457618i\)
\(L(1)\) \(\approx\) \(0.6913937367 + 0.2058457618i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
191 \( 1 \)
good2 \( 1 + (-0.965 + 0.261i)T \)
5 \( 1 + (-0.401 + 0.915i)T \)
11 \( 1 + (0.0825 + 0.996i)T \)
13 \( 1 + (0.574 - 0.818i)T \)
17 \( 1 + (0.431 + 0.901i)T \)
19 \( 1 + (0.148 + 0.988i)T \)
23 \( 1 + (0.461 - 0.887i)T \)
29 \( 1 + (-0.980 + 0.197i)T \)
31 \( 1 + (-0.789 - 0.614i)T \)
37 \( 1 + (0.789 - 0.614i)T \)
41 \( 1 + (0.546 - 0.837i)T \)
43 \( 1 + (0.371 - 0.928i)T \)
47 \( 1 + (-0.518 + 0.854i)T \)
53 \( 1 + (0.973 + 0.229i)T \)
59 \( 1 + (0.828 - 0.560i)T \)
61 \( 1 + (0.724 + 0.689i)T \)
67 \( 1 + (0.991 + 0.131i)T \)
71 \( 1 + (0.934 - 0.355i)T \)
73 \( 1 + (-0.652 + 0.757i)T \)
79 \( 1 + (0.980 + 0.197i)T \)
83 \( 1 + (-0.461 - 0.887i)T \)
89 \( 1 + (-0.768 + 0.639i)T \)
97 \( 1 + (0.277 - 0.960i)T \)
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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

−18.39842164794222478261511633039, −17.81130320890219111361455305878, −16.76908363032163684638617457186, −16.566803961406027813481155331765, −15.94130905974542673685583700212, −15.30426317967910192224401223304, −14.3030000342265045296390602045, −13.29222421965857659503851133119, −12.98303907709723166102797103980, −11.835377490346897843456245580490, −11.39495390844407680951402263363, −11.08347690234466569363499896946, −9.80307123993171087344173847381, −9.25610958973153063514686553077, −8.79374033214425986206207550324, −8.05463177025909770464409352473, −7.36776567852439482745897962898, −6.624493027893750427551658729754, −5.69270801769352113062956403692, −4.926596800526534075761287409969, −3.83222364329043481215437947692, −3.28344637291201664906439674349, −2.26154221511643267547047399023, −1.22682218640427499913162366319, −0.698961619534089275345243505234, 0.67569531478302783745456164884, 1.81792074424044239819191556013, 2.44743520589600804422989790856, 3.47674915465768175787883311664, 4.10676055579421453227121231351, 5.531879528785383684796794445510, 5.95490246230702273915689038489, 6.893357207330734418766571281450, 7.453832212790879245974386785, 8.02383856336330816658230970753, 8.755064726952395430058180133878, 9.71794398031199726341634548142, 10.253051970670782417960881662180, 10.854910023554409321518086383451, 11.40105046755919440177969761810, 12.41586166812004302159528541236, 12.84032954582361832337569047193, 14.239860679452806080653152766018, 14.70555430680460568310254769437, 15.19044552245144593917621574720, 15.86156302892416849930648145393, 16.65023626972015407970735788427, 17.27140938607991294418389062177, 18.09692550595523562550993668593, 18.417385603522320957716891527585

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