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Properties

Label 1-105-105.17-r1-0-0
Degree $1$
Conductor $105$
Sign $-0.156 + 0.987i$
Analytic cond. $11.2838$
Root an. cond. $11.2838$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.156 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.156 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $-0.156 + 0.987i$
Analytic conductor: \(11.2838\)
Root analytic conductor: \(11.2838\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{105} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 105,\ (1:\ ),\ -0.156 + 0.987i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.811389767 + 2.120105693i\)
\(L(\frac12)\) \(\approx\) \(1.811389767 + 2.120105693i\)
\(L(1)\) \(\approx\) \(1.558988302 + 0.8671390436i\)
\(L(1)\) \(\approx\) \(1.558988302 + 0.8671390436i\)

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

−29.65162769126381317801274013190, −28.1711248653568268107540459891, −27.51279374403017242299167233018, −25.938285959553782568027316106607, −24.77700019899877822262600810081, −23.901151945483155487532745908009, −22.85279667960204528940062901071, −21.87356702915152994372663677151, −21.03774849337250959819659031050, −19.80112472080749755482284656710, −19.08716537654744831637364527077, −17.61784432691752959817640987837, −16.18914085650522333433797709483, −15.119154594873304106534882632861, −14.05444977457702051226449324128, −13.04730583625022113026095806874, −11.940327208715621195576407937241, −10.869241114657138087503112804798, −9.77315740806023969448895588, −8.13770839572759376426490657176, −6.45733475462387093109276529267, −5.43547563572545142456145900971, −3.962061405869019912897533012534, −2.763489543264713663393180448041, −0.97152758335965067345692074834, 2.034247183860731884092781772548, 3.73164202577169353113149144993, 4.83431089499032463264806659654, 6.26738410863518089795006908046, 7.27801141727382561322340482392, 8.613216779714191651174558326139, 10.16653743773725181359665410326, 11.82649685111012507425664641583, 12.44002784875433066084308874638, 13.98144223835120123844049983909, 14.579262474619503434295432482655, 15.922607923618600043139464964, 16.784826766044178160207265917338, 17.93747792327814077191974012194, 19.42611725196674377781380428766, 20.68541964978492134831108076776, 21.50320556327192892116736238898, 22.71825707983812011733009545939, 23.397069710082722830759158485108, 24.587040105164494436134299761983, 25.43372122304953932068851038632, 26.37627833044666502072135912074, 27.62914842289857164076855848802, 28.93765441335301439070310883678, 29.99254568077376053088559787444

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