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Properties

Label 1-7-7.4-r0-0-0
Degree $1$
Conductor $7$
Sign $0.895 + 0.444i$
Analytic cond. $0.0325078$
Root an. cond. $0.0325078$
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  + (−0.5 + 0.866i)2-s + (−0.5 − 0.866i)3-s + (−0.5 − 0.866i)4-s + (−0.5 + 0.866i)5-s + 6-s + 8-s + (−0.5 + 0.866i)9-s + (−0.5 − 0.866i)10-s + (−0.5 − 0.866i)11-s + (−0.5 + 0.866i)12-s + 13-s + 15-s + (−0.5 + 0.866i)16-s + (−0.5 − 0.866i)17-s + (−0.5 − 0.866i)18-s + (−0.5 + 0.866i)19-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)2-s + (−0.5 − 0.866i)3-s + (−0.5 − 0.866i)4-s + (−0.5 + 0.866i)5-s + 6-s + 8-s + (−0.5 + 0.866i)9-s + (−0.5 − 0.866i)10-s + (−0.5 − 0.866i)11-s + (−0.5 + 0.866i)12-s + 13-s + 15-s + (−0.5 + 0.866i)16-s + (−0.5 − 0.866i)17-s + (−0.5 − 0.866i)18-s + (−0.5 + 0.866i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.895 + 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.895 + 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(7\)
Sign: $0.895 + 0.444i$
Analytic conductor: \(0.0325078\)
Root analytic conductor: \(0.0325078\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{7} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 7,\ (0:\ ),\ 0.895 + 0.444i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3100893625 + 0.07264193137i\)
\(L(\frac12)\) \(\approx\) \(0.3100893625 + 0.07264193137i\)
\(L(1)\) \(\approx\) \(0.5377473805 + 0.1052975456i\)
\(L(1)\) \(\approx\) \(0.5377473805 + 0.1052975456i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
good2 \( 1 + (-0.5 + 0.866i)T \)
3 \( 1 + (-0.5 - 0.866i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
11 \( 1 + (-0.5 - 0.866i)T \)
13 \( 1 + T \)
17 \( 1 + (-0.5 - 0.866i)T \)
19 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + (-0.5 + 0.866i)T \)
29 \( 1 + T \)
31 \( 1 + (-0.5 - 0.866i)T \)
37 \( 1 + (-0.5 + 0.866i)T \)
41 \( 1 + T \)
43 \( 1 + T \)
47 \( 1 + (-0.5 + 0.866i)T \)
53 \( 1 + (-0.5 - 0.866i)T \)
59 \( 1 + (-0.5 - 0.866i)T \)
61 \( 1 + (-0.5 + 0.866i)T \)
67 \( 1 + (-0.5 - 0.866i)T \)
71 \( 1 + T \)
73 \( 1 + (-0.5 - 0.866i)T \)
79 \( 1 + (-0.5 + 0.866i)T \)
83 \( 1 + T \)
89 \( 1 + (-0.5 + 0.866i)T \)
97 \( 1 + T \)
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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

−50.62524764170310491440020688393, −49.1264751496983520934989031681, −47.49155924034895036487974399661, −46.096098653016577564786969576350, −44.59577153477944612483322514215, −43.539481156902262050279499562673, −40.476471625192229604298983514113, −39.33848314648563594220565047551, −38.20675542517363468993511520277, −36.34475584680101927414209976127, −34.79250339555226619207019919603, −32.61008859431403864338197930246, −30.91956093265655588210565187266, −28.59979377444089598980162089082, −27.87337549022489308550549301123, −26.16994490801983565967242517629, −23.20367246134665537826174805893, −21.31464724410425595182027902594, −20.03055898508203028994206564551, −17.61605319887654241030080166645, −15.93744820468795955688957399890, −12.53254782268627400807230480038, −10.73611998749339311587424153504, −8.78555471449907536558015746317, −4.35640162473628422727957479051, 6.20123004275588129466099054628, 7.92743089809203774838798659746, 11.010444862072490422393627410948, 13.82986789986136757061236809479, 16.01372713415040781987211528577, 18.04485754217402476822077016067, 19.113885719489582461848208597857, 22.75640595577430793123629559665, 23.955938435167978513930764480420, 25.72310440610835748550521669187, 27.45559608897488330169144708954, 29.338505180723688436571601984894, 31.284264524394350318670300126769, 33.672299474642688101357196188180, 34.77419497072559204614894177206, 35.973149707308886772627581763109, 37.7869208124192804866845011160, 40.22456635882812767541409095300, 41.90913793196379644129443420935, 42.71263117169095738635373004950, 44.97719959096264445682102434355, 46.08677439078507722936634790159, 47.3488008532672270198897618227, 50.01732628524154217273296113486, 51.267236570557977166074479101533

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