Properties

Label 1-175-175.137-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: $\chi_{175} (137, \cdot )$
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

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

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