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Properties

Label 1-4011-4011.1301-r1-0-0
Degree $1$
Conductor $4011$
Sign $0.706 + 0.707i$
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  + (−0.789 − 0.614i)2-s + (0.245 + 0.969i)4-s + (0.245 + 0.969i)5-s + (0.401 − 0.915i)8-s + (0.401 − 0.915i)10-s + (0.546 + 0.837i)11-s + (0.677 + 0.735i)13-s + (−0.879 + 0.475i)16-s + (0.945 + 0.324i)17-s + (−0.401 − 0.915i)19-s + (−0.879 + 0.475i)20-s + (0.0825 − 0.996i)22-s + (0.401 + 0.915i)23-s + (−0.879 + 0.475i)25-s + (−0.0825 − 0.996i)26-s + ⋯
L(s)  = 1  + (−0.789 − 0.614i)2-s + (0.245 + 0.969i)4-s + (0.245 + 0.969i)5-s + (0.401 − 0.915i)8-s + (0.401 − 0.915i)10-s + (0.546 + 0.837i)11-s + (0.677 + 0.735i)13-s + (−0.879 + 0.475i)16-s + (0.945 + 0.324i)17-s + (−0.401 − 0.915i)19-s + (−0.879 + 0.475i)20-s + (0.0825 − 0.996i)22-s + (0.401 + 0.915i)23-s + (−0.879 + 0.475i)25-s + (−0.0825 − 0.996i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.706 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.706 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4011\)    =    \(3 \cdot 7 \cdot 191\)
Sign: $0.706 + 0.707i$
Analytic conductor: \(431.041\)
Root analytic conductor: \(431.041\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4011} (1301, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4011,\ (1:\ ),\ 0.706 + 0.707i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.670569032 + 0.6929790394i\)
\(L(\frac12)\) \(\approx\) \(1.670569032 + 0.6929790394i\)
\(L(1)\) \(\approx\) \(0.8691353804 + 0.05206501776i\)
\(L(1)\) \(\approx\) \(0.8691353804 + 0.05206501776i\)

Euler product

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

−18.291362329751685756954028776523, −17.351961769619168282212292806602, −16.77353688470263781467160226556, −16.392342854215443702722038672154, −15.79638115525234084956781621841, −14.84638176052060399113242799788, −14.297369034956611058936733710, −13.542743928164695243278418044311, −12.80591114223814253691433329251, −11.986673778015480749045543759588, −11.23136164270763113044679594285, −10.37381620818702810694667352823, −9.87683202881398627037464110844, −8.976055297941399811243612103504, −8.39885502830433197068729205779, −8.10743299790956849283199239194, −6.97521090343549334206268316693, −6.27157826149122141752995958296, −5.497401048560463689734020309787, −5.13015248507555780087001645605, −3.95452404115422310189083329954, −3.0903911078356533437901220844, −1.80984186261325840245816952942, −1.12633424478429972689379008259, −0.52588278825334243784035551983, 0.66868842684011473029876358905, 1.878339167110495152125532159836, 2.06229462410779590703332411016, 3.3664847231001634169694420407, 3.68652309434391751945607669458, 4.69621070072185493329312175959, 5.9112633420037486943840442159, 6.682314082946569084008412330661, 7.22721491831801892720720092329, 7.91320254982774744969986815478, 8.84953018139269402715735386734, 9.62060422862839836202177083265, 9.90487401753851078721418102687, 10.91151318616411038471120212800, 11.37950856889442141588263051837, 11.902986768509277563982643448657, 12.928423720889831512246109764095, 13.44279979784377750279971958111, 14.32242018244184988163141369629, 15.1140747482013800127958859839, 15.60219549401764695727985097880, 16.83512241696453607305911148314, 16.98721626223087563984679190891, 17.911354091017149662835671093483, 18.402110749904450941290787593904

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