Properties

Label 1-175-175.142-r1-0-0
Degree $1$
Conductor $175$
Sign $-0.186 - 0.982i$
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  + (−0.406 + 0.913i)2-s + (0.743 − 0.669i)3-s + (−0.669 − 0.743i)4-s + (0.309 + 0.951i)6-s + (0.951 − 0.309i)8-s + (0.104 − 0.994i)9-s + (−0.104 − 0.994i)11-s + (−0.994 − 0.104i)12-s + (−0.587 + 0.809i)13-s + (−0.104 + 0.994i)16-s + (0.207 − 0.978i)17-s + (0.866 + 0.5i)18-s + (−0.669 + 0.743i)19-s + (0.951 + 0.309i)22-s + (0.406 − 0.913i)23-s + (0.5 − 0.866i)24-s + ⋯
L(s)  = 1  + (−0.406 + 0.913i)2-s + (0.743 − 0.669i)3-s + (−0.669 − 0.743i)4-s + (0.309 + 0.951i)6-s + (0.951 − 0.309i)8-s + (0.104 − 0.994i)9-s + (−0.104 − 0.994i)11-s + (−0.994 − 0.104i)12-s + (−0.587 + 0.809i)13-s + (−0.104 + 0.994i)16-s + (0.207 − 0.978i)17-s + (0.866 + 0.5i)18-s + (−0.669 + 0.743i)19-s + (0.951 + 0.309i)22-s + (0.406 − 0.913i)23-s + (0.5 − 0.866i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.186 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.186 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(175\)    =    \(5^{2} \cdot 7\)
Sign: $-0.186 - 0.982i$
Analytic conductor: \(18.8063\)
Root analytic conductor: \(18.8063\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{175} (142, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 175,\ (1:\ ),\ -0.186 - 0.982i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6538322312 - 0.7899103657i\)
\(L(\frac12)\) \(\approx\) \(0.6538322312 - 0.7899103657i\)
\(L(1)\) \(\approx\) \(0.8935952148 - 0.06717361359i\)
\(L(1)\) \(\approx\) \(0.8935952148 - 0.06717361359i\)

Euler product

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

−27.75919294016148995200452106334, −26.55961094470538668485980428872, −25.84579799195837014593698591730, −25.02195178169513791951061703819, −23.362572239865492991968070945908, −22.258551749820986827392096127698, −21.49819244695072775116217798623, −20.586337675769529755681269876881, −19.78180865645053446233054687973, −19.11228441922736092489971135927, −17.7004475130860413584342806718, −16.963595024625216600999143847975, −15.48209100612524456369216395910, −14.67700998159788770242745987845, −13.31196799693790195377422231388, −12.5651309937074872433339348116, −11.12338645298296823395965144497, −10.173696258988092890314325939093, −9.445187579635081110726880129792, −8.3202801883246645327607954391, −7.38501640003638303203419848618, −5.129239135743421744805725155517, −4.07320172606015371175246145465, −2.91551893787258639907955556794, −1.78741071375725809233536895305, 0.37169793690431145388277935216, 1.94656533015973018280227519375, 3.64647132906890061909908103654, 5.24385664241092621124835092991, 6.57461332113919253133329168393, 7.35543072375411237871813169022, 8.53375789126583170071192436191, 9.16912229365828419400603449602, 10.51160645234708295097132820876, 12.091844429304983435169544792808, 13.358685953522978911367682707271, 14.198808104524216228891313183162, 14.891581917294405153217160659, 16.23067279335771755016236340730, 17.00690601098902644834300747827, 18.44130358402204296601266677370, 18.793523645840140874148627292669, 19.81666992716180886009706346310, 21.00991119881428401197590576162, 22.34051401711847255452435498630, 23.563929770698139546008996610157, 24.23649262779876432042432808317, 25.00890592977087929632138184548, 25.89043671622410890419754834426, 26.75652258058110990818388365330

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