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Properties

Label 1-4011-4011.41-r1-0-0
Degree $1$
Conductor $4011$
Sign $0.368 - 0.929i$
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  + (−0.245 − 0.969i)2-s + (−0.879 + 0.475i)4-s + (−0.879 + 0.475i)5-s + (0.677 + 0.735i)8-s + (0.677 + 0.735i)10-s + (−0.401 + 0.915i)11-s + (0.0825 − 0.996i)13-s + (0.546 − 0.837i)16-s + (0.789 + 0.614i)17-s + (−0.677 + 0.735i)19-s + (0.546 − 0.837i)20-s + (0.986 + 0.164i)22-s + (0.677 − 0.735i)23-s + (0.546 − 0.837i)25-s + (−0.986 + 0.164i)26-s + ⋯
L(s)  = 1  + (−0.245 − 0.969i)2-s + (−0.879 + 0.475i)4-s + (−0.879 + 0.475i)5-s + (0.677 + 0.735i)8-s + (0.677 + 0.735i)10-s + (−0.401 + 0.915i)11-s + (0.0825 − 0.996i)13-s + (0.546 − 0.837i)16-s + (0.789 + 0.614i)17-s + (−0.677 + 0.735i)19-s + (0.546 − 0.837i)20-s + (0.986 + 0.164i)22-s + (0.677 − 0.735i)23-s + (0.546 − 0.837i)25-s + (−0.986 + 0.164i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.368 - 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.368 - 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4011\)    =    \(3 \cdot 7 \cdot 191\)
Sign: $0.368 - 0.929i$
Analytic conductor: \(431.041\)
Root analytic conductor: \(431.041\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4011} (41, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4011,\ (1:\ ),\ 0.368 - 0.929i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9181255417 - 0.6237675637i\)
\(L(\frac12)\) \(\approx\) \(0.9181255417 - 0.6237675637i\)
\(L(1)\) \(\approx\) \(0.6991806758 - 0.2108528420i\)
\(L(1)\) \(\approx\) \(0.6991806758 - 0.2108528420i\)

Euler product

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

−18.56749853670264856486632150362, −17.53171817345572888753746701435, −16.96891390684690276041819601420, −16.21706878525911111283688271791, −15.95138240579743815774507488659, −15.200717486632671844618449774307, −14.44443740002104717558520565762, −13.8020641180786407380821352893, −13.09082722220817226617637199260, −12.433345574785965750088139471782, −11.333448368660373549245220956508, −11.07072358170387537440810434263, −9.86911010341392326278796008358, −9.1492277050541036166554067042, −8.66980815129245182098590022526, −7.877053677526491440210284792243, −7.34828962801005641819000678092, −6.60131499396341264449637068129, −5.69753076362263238399986775518, −5.082732632020965606220871605513, −4.264069210275136402098257946782, −3.6788181253339809097941625669, −2.57911083214303680867871885063, −1.182541065623476122056700293153, −0.545317789822358811803576517129, 0.35484764917104918409477378976, 1.28493540634199496647626440351, 2.27301358053919531420182272602, 3.062540887951566525846281823815, 3.64540510550057108311193112873, 4.46796802299973148849207910597, 5.13796678890587976664267580000, 6.18278635833083832177348558345, 7.24652388208590938571325513131, 7.87923539271065950210783627389, 8.340555923051561508424672088527, 9.26095642198868623191186477360, 10.18934226992731620784262943165, 10.61359059492424149163208023531, 11.08328579581264837248463228987, 12.23956322933777066761706861190, 12.44903374540523367418238224708, 13.023373084506990033057037633914, 14.12936291222001638950161229484, 14.84744337169541273307715814770, 15.20215921447114821746504910076, 16.3278256006609399092029539958, 16.88549930890373847371902030402, 17.764223258263459056378072283527, 18.39451319209635934367476879003

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