Properties

Label 1-105-105.53-r0-0-0
Degree $1$
Conductor $105$
Sign $0.995 - 0.0932i$
Analytic cond. $0.487617$
Root an. cond. $0.487617$
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  + (−0.866 − 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)  = 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

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

Invariants

Degree: \(1\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $0.995 - 0.0932i$
Analytic conductor: \(0.487617\)
Root analytic conductor: \(0.487617\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{105} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 105,\ (0:\ ),\ 0.995 - 0.0932i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7245183557 - 0.03384175778i\)
\(L(\frac12)\) \(\approx\) \(0.7245183557 - 0.03384175778i\)
\(L(1)\) \(\approx\) \(0.7576548819 - 0.07122149674i\)
\(L(1)\) \(\approx\) \(0.7576548819 - 0.07122149674i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good2 \( 1 + (0.866 + 0.5i)T \)
11 \( 1 + (-0.5 - 0.866i)T \)
13 \( 1 - iT \)
17 \( 1 + (-0.866 + 0.5i)T \)
19 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + (-0.866 - 0.5i)T \)
29 \( 1 - T \)
31 \( 1 + (0.5 + 0.866i)T \)
37 \( 1 + (-0.866 - 0.5i)T \)
41 \( 1 + T \)
43 \( 1 - iT \)
47 \( 1 + (0.866 + 0.5i)T \)
53 \( 1 + (0.866 - 0.5i)T \)
59 \( 1 + (0.5 + 0.866i)T \)
61 \( 1 + (0.5 - 0.866i)T \)
67 \( 1 + (0.866 - 0.5i)T \)
71 \( 1 + T \)
73 \( 1 + (-0.866 + 0.5i)T \)
79 \( 1 + (-0.5 + 0.866i)T \)
83 \( 1 + iT \)
89 \( 1 + (0.5 - 0.866i)T \)
97 \( 1 + iT \)
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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

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

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