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Properties

Label 1-4011-4011.2876-r1-0-0
Degree $1$
Conductor $4011$
Sign $-0.889 - 0.456i$
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  + (0.401 + 0.915i)2-s + (−0.677 + 0.735i)4-s + (−0.677 + 0.735i)5-s + (−0.945 − 0.324i)8-s + (−0.945 − 0.324i)10-s + (−0.986 − 0.164i)11-s + (−0.789 − 0.614i)13-s + (−0.0825 − 0.996i)16-s + (0.546 − 0.837i)17-s + (0.945 − 0.324i)19-s + (−0.0825 − 0.996i)20-s + (−0.245 − 0.969i)22-s + (−0.945 + 0.324i)23-s + (−0.0825 − 0.996i)25-s + (0.245 − 0.969i)26-s + ⋯
L(s)  = 1  + (0.401 + 0.915i)2-s + (−0.677 + 0.735i)4-s + (−0.677 + 0.735i)5-s + (−0.945 − 0.324i)8-s + (−0.945 − 0.324i)10-s + (−0.986 − 0.164i)11-s + (−0.789 − 0.614i)13-s + (−0.0825 − 0.996i)16-s + (0.546 − 0.837i)17-s + (0.945 − 0.324i)19-s + (−0.0825 − 0.996i)20-s + (−0.245 − 0.969i)22-s + (−0.945 + 0.324i)23-s + (−0.0825 − 0.996i)25-s + (0.245 − 0.969i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.889 - 0.456i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.889 - 0.456i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4011\)    =    \(3 \cdot 7 \cdot 191\)
Sign: $-0.889 - 0.456i$
Analytic conductor: \(431.041\)
Root analytic conductor: \(431.041\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4011} (2876, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4011,\ (1:\ ),\ -0.889 - 0.456i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.04197544009 + 0.1737869164i\)
\(L(\frac12)\) \(\approx\) \(-0.04197544009 + 0.1737869164i\)
\(L(1)\) \(\approx\) \(0.7318027253 + 0.3891302398i\)
\(L(1)\) \(\approx\) \(0.7318027253 + 0.3891302398i\)

Euler product

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

−18.145536023457022121787332735353, −17.2795126375512211332016049826, −16.5520515513756309348317775819, −15.76230918260044921631625151013, −15.16781319734731086758192366729, −14.3715139807731773150351716847, −13.70316706699675060312981430986, −12.949066707397055298160606534314, −12.31103081998008911749082871687, −11.907805244361438830859951877263, −11.22102986962805185700439381235, −10.19727344733828440942205545413, −9.883068352791241509566568295441, −8.99529471747114385935549980518, −8.111376750329936854534199790942, −7.720330261510735002423062300445, −6.486472589994619349432947964293, −5.57948900252100580830609583559, −4.91567085501177966646257280840, −4.3577775825419090392764255052, −3.547235726497441626633368896527, −2.77460957616125697865795560139, −1.836718257047959953673887410934, −1.08119298217464186433608776388, −0.039210641857380434314987634032, 0.577606541894653636652787091534, 2.38266721748682764425400100875, 3.03933354130508058349180309606, 3.67135509690486916159046163506, 4.61135250835141037460720158014, 5.40016442454878414869307758660, 5.84323080932034511668415897426, 7.096992638446328847461175201405, 7.37070777884865457044010473724, 7.91616102961259691100551935333, 8.738762232364997591169015539190, 9.74553524182551833133501906484, 10.27902580800775508014316343221, 11.31024873404329504532482035763, 11.94792980281579271867501342713, 12.61211968062213280459988014325, 13.38070530072695342868174393209, 14.19236172742361344979073435247, 14.53972509412768724285969550249, 15.47863140223639041479623685520, 15.85300238499735312245125298989, 16.33808254078959664378838689383, 17.43888286682533796475212107699, 17.89558349642394780000226998562, 18.604590629680172475907907392497

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