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Properties

Label 1-175-175.167-r0-0-0
Degree $1$
Conductor $175$
Sign $0.968 - 0.248i$
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  + (−0.587 − 0.809i)2-s + (0.951 + 0.309i)3-s + (−0.309 + 0.951i)4-s + (−0.309 − 0.951i)6-s + (0.951 − 0.309i)8-s + (0.809 + 0.587i)9-s + (−0.809 + 0.587i)11-s + (−0.587 + 0.809i)12-s + (0.587 − 0.809i)13-s + (−0.809 − 0.587i)16-s + (0.951 − 0.309i)17-s − i·18-s + (0.309 + 0.951i)19-s + (0.951 + 0.309i)22-s + (0.587 + 0.809i)23-s + 24-s + ⋯
L(s)  = 1  + (−0.587 − 0.809i)2-s + (0.951 + 0.309i)3-s + (−0.309 + 0.951i)4-s + (−0.309 − 0.951i)6-s + (0.951 − 0.309i)8-s + (0.809 + 0.587i)9-s + (−0.809 + 0.587i)11-s + (−0.587 + 0.809i)12-s + (0.587 − 0.809i)13-s + (−0.809 − 0.587i)16-s + (0.951 − 0.309i)17-s − i·18-s + (0.309 + 0.951i)19-s + (0.951 + 0.309i)22-s + (0.587 + 0.809i)23-s + 24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.968 - 0.248i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.968 - 0.248i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(175\)    =    \(5^{2} \cdot 7\)
Sign: $0.968 - 0.248i$
Analytic conductor: \(0.812696\)
Root analytic conductor: \(0.812696\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{175} (167, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 175,\ (0:\ ),\ 0.968 - 0.248i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.143618972 - 0.1444726740i\)
\(L(\frac12)\) \(\approx\) \(1.143618972 - 0.1444726740i\)
\(L(1)\) \(\approx\) \(1.053928011 - 0.1647003641i\)
\(L(1)\) \(\approx\) \(1.053928011 - 0.1647003641i\)

Euler product

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

−27.10683614100709712929315416378, −26.2705725464040506073960032357, −25.77982604830654761241281640435, −24.67022163796036914461444257819, −23.9539674635853278845091690086, −23.13105164169777042887912750742, −21.49516761544082471805390424153, −20.56408963281344321199153956278, −19.3260279654351595278549383980, −18.77176977427090345765031696274, −17.87804118536774177153757017744, −16.52329310098867726334022042312, −15.70456185618224061995157121733, −14.66667273697660453153936166074, −13.82496685635903737653473142544, −12.92882605029647681166275405278, −11.1672765403875267574158409913, −9.9320134089896622605204255383, −8.91334946042863765740124819687, −8.11210938369060198459988167159, −7.123405907419665460998080200352, −6.00079241214806202797776462853, −4.50721941682342224370671121917, −2.87578778823018918533931336313, −1.27683940519418989366681592498, 1.552293778148158696344819229541, 2.892068689648991385967030042710, 3.75890017065205525674049303399, 5.236158785380861030521143346628, 7.49601488528103081345836454894, 8.04282527034863761419716564162, 9.32481628689839218717268802317, 10.08013478330780188947738843573, 11.0418120094577692273289297037, 12.547762328078563286164555162662, 13.24683227572971523278472178069, 14.457605138259856081341589605545, 15.66213423250599681414085200566, 16.61294889924377955510459138443, 18.0475587476875433817382142905, 18.69335339306112304721751487786, 19.774731035212308468782469271164, 20.674028136240082124172627280699, 21.08358609992845649090820950978, 22.33529997799660514867375354251, 23.38909815546936281578505520282, 25.12269249964410558896222020390, 25.57944800300489431624542111090, 26.54060717122595435069714411984, 27.46814992144664376307831482830

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