Properties

Label 1-175-175.173-r0-0-0
Degree $1$
Conductor $175$
Sign $0.743 + 0.668i$
Analytic cond. $0.812696$
Root an. cond. $0.812696$
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  + (0.207 + 0.978i)2-s + (−0.406 − 0.913i)3-s + (−0.913 + 0.406i)4-s + (0.809 − 0.587i)6-s + (−0.587 − 0.809i)8-s + (−0.669 + 0.743i)9-s + (0.669 + 0.743i)11-s + (0.743 + 0.669i)12-s + (0.951 − 0.309i)13-s + (0.669 − 0.743i)16-s + (0.994 − 0.104i)17-s + (−0.866 − 0.5i)18-s + (0.913 + 0.406i)19-s + (−0.587 + 0.809i)22-s + (−0.207 − 0.978i)23-s + (−0.5 + 0.866i)24-s + ⋯
L(s)  = 1  + (0.207 + 0.978i)2-s + (−0.406 − 0.913i)3-s + (−0.913 + 0.406i)4-s + (0.809 − 0.587i)6-s + (−0.587 − 0.809i)8-s + (−0.669 + 0.743i)9-s + (0.669 + 0.743i)11-s + (0.743 + 0.669i)12-s + (0.951 − 0.309i)13-s + (0.669 − 0.743i)16-s + (0.994 − 0.104i)17-s + (−0.866 − 0.5i)18-s + (0.913 + 0.406i)19-s + (−0.587 + 0.809i)22-s + (−0.207 − 0.978i)23-s + (−0.5 + 0.866i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.743 + 0.668i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.743 + 0.668i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(175\)    =    \(5^{2} \cdot 7\)
Sign: $0.743 + 0.668i$
Analytic conductor: \(0.812696\)
Root analytic conductor: \(0.812696\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{175} (173, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 175,\ (0:\ ),\ 0.743 + 0.668i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9797356834 + 0.3754746417i\)
\(L(\frac12)\) \(\approx\) \(0.9797356834 + 0.3754746417i\)
\(L(1)\) \(\approx\) \(0.9644375316 + 0.2738050804i\)
\(L(1)\) \(\approx\) \(0.9644375316 + 0.2738050804i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
good2 \( 1 + (0.207 + 0.978i)T \)
3 \( 1 + (-0.406 - 0.913i)T \)
11 \( 1 + (0.669 + 0.743i)T \)
13 \( 1 + (0.951 - 0.309i)T \)
17 \( 1 + (0.994 - 0.104i)T \)
19 \( 1 + (0.913 + 0.406i)T \)
23 \( 1 + (-0.207 - 0.978i)T \)
29 \( 1 + (0.809 + 0.587i)T \)
31 \( 1 + (0.104 + 0.994i)T \)
37 \( 1 + (-0.743 - 0.669i)T \)
41 \( 1 + (-0.309 - 0.951i)T \)
43 \( 1 + iT \)
47 \( 1 + (-0.994 - 0.104i)T \)
53 \( 1 + (0.406 + 0.913i)T \)
59 \( 1 + (-0.978 - 0.207i)T \)
61 \( 1 + (0.978 - 0.207i)T \)
67 \( 1 + (-0.994 + 0.104i)T \)
71 \( 1 + (-0.809 - 0.587i)T \)
73 \( 1 + (0.743 - 0.669i)T \)
79 \( 1 + (0.104 - 0.994i)T \)
83 \( 1 + (0.587 + 0.809i)T \)
89 \( 1 + (-0.978 + 0.207i)T \)
97 \( 1 + (0.587 - 0.809i)T \)
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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

−27.55989639009324196148622589682, −26.72799824608092510498775290082, −25.7281750325613110959411481324, −24.08213390232688284064118614590, −23.15836932776986208637314951544, −22.31058195949054914078662519519, −21.46204900896723960224692903372, −20.79938059248118932146659227030, −19.74873063981090145799540530390, −18.71730737556844110984952903520, −17.60942011524698062134216143039, −16.58050705137990438250454521295, −15.46608625588693029364927186819, −14.264762549884324171948309141656, −13.446248020562053105632907738358, −11.79494646582711032075991741034, −11.46070905129206708164386830559, −10.22225656838285018031429562739, −9.39899587898599298996893135550, −8.35964321649641854043577440104, −6.1878312021091750126474680406, −5.23399533252627905281514799607, −3.928526042596736706191642950937, −3.17509459750197425546593742920, −1.153922591516583840028484871834, 1.25579582554016724101579035613, 3.33208613500128635950758953479, 4.89924841055472443892533544472, 5.98074755432580654932757996615, 6.89051128254044758069299057910, 7.84820537560086676480917529368, 8.87884323516486944446832341981, 10.348161915998625828259974531120, 11.992306431937068742657561619716, 12.60980367436480750207658098755, 13.84966614422364655579423878851, 14.49305086103660622253177728428, 15.936459521831631197985218455478, 16.7326415128704686840507667258, 17.84029393272852577300893006295, 18.34606343788864219580842879292, 19.54181697260883308687219056757, 20.87880637136145921763750186236, 22.32267627014309838709104315791, 22.96051416868312839330105394139, 23.67316083671286632385706196042, 24.86428322398254778141164955264, 25.23467110904049414189483749197, 26.31226248080069941424487355309, 27.62888723248432350184982599245

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