Properties

Degree 1
Conductor $ 3 \cdot 7 \cdot 191 $
Sign $0.368 - 0.929i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,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,χ)  = 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

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

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(4011\)    =    \(3 \cdot 7 \cdot 191\)
\( \varepsilon \)  =  $0.368 - 0.929i$
motivic weight  =  \(0\)
character  :  $\chi_{4011} (41, \cdot )$
Sato-Tate  :  $\mu(38)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 4011,\ (1:\ ),\ 0.368 - 0.929i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.9181255417 - 0.6237675637i$
$L(\frac12,\chi)$  $\approx$  $0.9181255417 - 0.6237675637i$
$L(\chi,1)$  $\approx$  0.6991806758 - 0.2108528420i
$L(1,\chi)$  $\approx$  0.6991806758 - 0.2108528420i

Euler product

\[\begin{aligned} L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]
\[\begin{aligned} L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]

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