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Properties

Label 1-175-175.159-r1-0-0
Degree $1$
Conductor $175$
Sign $-0.895 + 0.445i$
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  + (−0.669 + 0.743i)2-s + (−0.104 − 0.994i)3-s + (−0.104 − 0.994i)4-s + (0.809 + 0.587i)6-s + (0.809 + 0.587i)8-s + (−0.978 + 0.207i)9-s + (−0.978 − 0.207i)11-s + (−0.978 + 0.207i)12-s + (0.309 − 0.951i)13-s + (−0.978 + 0.207i)16-s + (0.913 − 0.406i)17-s + (0.5 − 0.866i)18-s + (0.104 − 0.994i)19-s + (0.809 − 0.587i)22-s + (−0.669 + 0.743i)23-s + (0.5 − 0.866i)24-s + ⋯
L(s)  = 1  + (−0.669 + 0.743i)2-s + (−0.104 − 0.994i)3-s + (−0.104 − 0.994i)4-s + (0.809 + 0.587i)6-s + (0.809 + 0.587i)8-s + (−0.978 + 0.207i)9-s + (−0.978 − 0.207i)11-s + (−0.978 + 0.207i)12-s + (0.309 − 0.951i)13-s + (−0.978 + 0.207i)16-s + (0.913 − 0.406i)17-s + (0.5 − 0.866i)18-s + (0.104 − 0.994i)19-s + (0.809 − 0.587i)22-s + (−0.669 + 0.743i)23-s + (0.5 − 0.866i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.895 + 0.445i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.895 + 0.445i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(175\)    =    \(5^{2} \cdot 7\)
Sign: $-0.895 + 0.445i$
Analytic conductor: \(18.8063\)
Root analytic conductor: \(18.8063\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{175} (159, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 175,\ (1:\ ),\ -0.895 + 0.445i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.003905453942 + 0.01661038660i\)
\(L(\frac12)\) \(\approx\) \(0.003905453942 + 0.01661038660i\)
\(L(1)\) \(\approx\) \(0.5619866546 - 0.05613915431i\)
\(L(1)\) \(\approx\) \(0.5619866546 - 0.05613915431i\)

Euler product

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

−27.857608636464383982097518273573, −26.84524741927482960038727960384, −26.143400876549738874108913108176, −25.41598489244868723559645993350, −23.70274026042626010354412547739, −22.65034276543597913038192713308, −21.65740616032703955892934812435, −20.88916638143172214228926344681, −20.2829221684066236989927409708, −18.931586057862289472898772549922, −18.18076939781754706212253482075, −16.76149498770785276243401076919, −16.39053973164023790123342787436, −15.072645377068411656372363114536, −13.821951529130119874634752282332, −12.46468170895510524843728937180, −11.48204716915458628419730748653, −10.44340546702902595953111200894, −9.784708762145056724058255350251, −8.66009471552691346457052278724, −7.6439812381786555559264212740, −5.84585989441703458882930891770, −4.38236685131911725715544173235, −3.41604948698422379010179515573, −1.98975011392363592858636355871, 0.00812522117665349552628259289, 1.29213132888960329382576326122, 2.87067022943826259967264363940, 5.22661600278973660318820997143, 5.95443331567032318878147166113, 7.35477847431505319460749034317, 7.88337214455221245865647916498, 9.059301158642494155003086217118, 10.41126654847663205127967279297, 11.42335850490603061352748476797, 12.905820807171797844798823132585, 13.71030781784502167831409079758, 14.871144999588272756478679987696, 15.95505181049114597978173404522, 16.954984819463370490673350625003, 18.13322828560768500247316149411, 18.37270076360334155576132475647, 19.63757333643328511726718386490, 20.405679819746633677390658451842, 22.109484850978790595464354406063, 23.47056530771503455678748854763, 23.65250008565444619790086897702, 24.922958014978633755381915967697, 25.54808757972001168234746557631, 26.41732726271650960395442935637

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