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Properties

Label 1-175-175.23-r1-0-0
Degree $1$
Conductor $175$
Sign $-0.124 - 0.992i$
Analytic cond. $18.8063$
Root an. cond. $18.8063$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.124 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.124 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(175\)    =    \(5^{2} \cdot 7\)
Sign: $-0.124 - 0.992i$
Analytic conductor: \(18.8063\)
Root analytic conductor: \(18.8063\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{175} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 175,\ (1:\ ),\ -0.124 - 0.992i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.5213247788 + 0.5910031381i\)
\(L(\frac12)\) \(\approx\) \(-0.5213247788 + 0.5910031381i\)
\(L(1)\) \(\approx\) \(0.5387505535 + 0.7673289226i\)
\(L(1)\) \(\approx\) \(0.5387505535 + 0.7673289226i\)

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

−26.76431890907719669619449289431, −25.44629469362884207059886990666, −24.40821017766496900860635697621, −23.63980729672403730828436577790, −22.52963931673840334102498000346, −21.64311572273823796701917480330, −20.509653926573094094357748217697, −19.46405738142114733992259703948, −19.19434811643538591478237332575, −17.85200548184796199072969174185, −17.17898609627691058642014533103, −15.25291414522867828861055020488, −14.16854055475011990818382035771, −13.49100529821877939983350879925, −12.387250674227741227784196432217, −11.69284668801037185530353722682, −10.45064465028576446517846437568, −9.12178590539579152848902526533, −8.32275002778998183281839924512, −6.8348362816027083233507093300, −5.57007958301688739990071594440, −4.002644280584112451515414621842, −2.78750707612231993241257099902, −1.67406252268411936264959356068, −0.241514861538629619834352461276, 2.53524782147920645516669445279, 4.22771110723759523826228345302, 4.67718226911953059938738603150, 6.22607621728052510687819872267, 7.353519511781748887045102035799, 8.66644950754158542147949143267, 9.372087813127571825180599124439, 10.50982289510809396909547484846, 12.08240711650319123861744601207, 13.28951904972506999863282905456, 14.536568294872431632461765080192, 14.93190033532527951618504944184, 16.02319830947856519500701848230, 16.95182050218055523693491547105, 17.72084852982085700721929850096, 19.24704877257008730115757751788, 20.18393396442163512323200384580, 21.4936586947960549068351874103, 22.16760104140051867197022406195, 22.986293473157123836902223650585, 24.28359968951329889383190928246, 25.07784603761573508498307477670, 26.05022054835608597105397595825, 26.71259720283291826831119016048, 27.61908620101163934422579151423

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