Properties

Degree 1
Conductor $ 3 \cdot 5 \cdot 7 $
Sign $0.995 + 0.0932i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.866 + 0.5i)2-s + (0.5 − 0.866i)4-s + i·8-s + (0.5 − 0.866i)11-s i·13-s + (−0.5 − 0.866i)16-s + (0.866 + 0.5i)17-s + (0.5 + 0.866i)19-s + i·22-s + (0.866 − 0.5i)23-s + (0.5 + 0.866i)26-s + 29-s + (−0.5 + 0.866i)31-s + (0.866 + 0.5i)32-s − 34-s + ⋯
L(s,χ)  = 1  + (−0.866 + 0.5i)2-s + (0.5 − 0.866i)4-s + i·8-s + (0.5 − 0.866i)11-s i·13-s + (−0.5 − 0.866i)16-s + (0.866 + 0.5i)17-s + (0.5 + 0.866i)19-s + i·22-s + (0.866 − 0.5i)23-s + (0.5 + 0.866i)26-s + 29-s + (−0.5 + 0.866i)31-s + (0.866 + 0.5i)32-s − 34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.995 + 0.0932i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 105 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.995 + 0.0932i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(105\)    =    \(3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $0.995 + 0.0932i$
motivic weight  =  \(0\)
character  :  $\chi_{105} (2, \cdot )$
Sato-Tate  :  $\mu(12)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 105,\ (0:\ ),\ 0.995 + 0.0932i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.7245183557 + 0.03384175778i$
$L(\frac12,\chi)$  $\approx$  $0.7245183557 + 0.03384175778i$
$L(\chi,1)$  $\approx$  0.7576548819 + 0.07122149674i
$L(1,\chi)$  $\approx$  0.7576548819 + 0.07122149674i

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

−29.52307112175659053127358579200, −28.59037487945380627964209707515, −27.71862002703768501841729225657, −26.7550832137935384163871315993, −25.721557887483261503833025603161, −24.91361033941547509528063330360, −23.513334180126688196391328656959, −22.1846197347678109666745806876, −21.18539624639958458011387346696, −20.16981696410980663618312019745, −19.22618639482586948998590025545, −18.20193458217397866778104516995, −17.16717813129192940903904541783, −16.23780958441153720636823281645, −14.92283190475603011232828219970, −13.422704495135924466958262428993, −12.08685444665869354861064790575, −11.312534087556372437868991241311, −9.83549612096686191013846472978, −9.12461780846155767598867902407, −7.637885440196233877385623629344, −6.64717638898275485498567875126, −4.586250112438853644447667121938, −3.016202078233119048861531401704, −1.47323464074645274625666854748, 1.19758382813300735391753483741, 3.19293580887404261068090611130, 5.308214054393170365763837959512, 6.373718433653193559247897545431, 7.77737889384323535001617498221, 8.69738344833147717050250361244, 10.01339171595318774215682726944, 10.96892090701662387114494545513, 12.34831430505831956525643310329, 14.00616695998076775804494791123, 14.960093579887217963849415224198, 16.18665565203927395756271287769, 16.998568297245835424286771940567, 18.14959623548031518225176802636, 19.097675552083183836941560181775, 20.0672947012960620364208660701, 21.242299842092793180847808790229, 22.70928856574814403799326967201, 23.743094486931191291655557800681, 24.89826443727250078963050328478, 25.47611825911802291192644348153, 26.94289898659468712667088893775, 27.32312555731768786650359891353, 28.60820850899327563625922261152, 29.4858983106391343056817902926

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