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Properties

Label 1-7-7.5-r1-0-0
Degree $1$
Conductor $7$
Sign $0.386 - 0.922i$
Analytic cond. $0.752254$
Root an. cond. $0.752254$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(7\)
Sign: $0.386 - 0.922i$
Analytic conductor: \(0.752254\)
Root analytic conductor: \(0.752254\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{7} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 7,\ (1:\ ),\ 0.386 - 0.922i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7139433437 - 0.4749021827i\)
\(L(\frac12)\) \(\approx\) \(0.7139433437 - 0.4749021827i\)
\(L(1)\) \(\approx\) \(0.8042057293 - 0.3986666988i\)
\(L(1)\) \(\approx\) \(0.8042057293 - 0.3986666988i\)

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

−50.97331252159187568512168359015, −49.543555603918569686782807799117, −47.758291071614958792796226880103, −45.46395163943458587355744436856, −44.259257194837827318219331433632, −43.330444112276133461209826986247, −41.63369016849716437682011847588, −39.57467223828350455921200958005, −37.37953167877961162723514730528, −36.447983987061088122490805917581, −34.35044122160486850148200883020, −32.747390632813393136386987652934, −31.84774663804095524857637113741, −28.64452753804391579441559340595, −27.13547137980929158540938389112, −25.68439458577475868571703403827, −24.15466453997877089700472248737, −21.65252506979642618329545373529, −19.651224233233595369541105291582, −17.161416543706070422905522561585, −15.74686940763941532761353888536, −13.85454287448149778875634224346, −9.97989590209139315060581291354, −8.41361099147117759845752355454, −5.19811619946654558608428407430, 2.50937455292911971967838452268, 7.48493173971596112913314844807, 9.89354379409772210349418069925, 12.25742488648921665489461478678, 14.13507775903777080989456447454, 17.714092561531158953226990374540, 18.88909760017588073794865307957, 20.60481911491253262583427068994, 22.66635642792466587252079667063, 25.28550752850252321309973718800, 26.66048203878250296742275725877, 28.99723231317671624580023005581, 30.13244058379747752560145687452, 31.44589058256335008426239173571, 34.31134814871948200646318610558, 36.070041076617265413000694622082, 37.18948618688047471474084289645, 38.58895062546931961023389784006, 40.69041710800720757960616041849, 41.99640907775237980637234247240, 44.16095204781287510286471187531, 45.95562251176600699194892072223, 46.96122667000339896653265574287, 48.57324539474628848818818146195, 49.51857626920139206120002047638

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