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Properties

Label 1-175-175.123-r1-0-0
Degree $1$
Conductor $175$
Sign $0.249 + 0.968i$
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  + (0.743 − 0.669i)2-s + (−0.994 − 0.104i)3-s + (0.104 − 0.994i)4-s + (−0.809 + 0.587i)6-s + (−0.587 − 0.809i)8-s + (0.978 + 0.207i)9-s + (−0.978 + 0.207i)11-s + (−0.207 + 0.978i)12-s + (−0.951 + 0.309i)13-s + (−0.978 − 0.207i)16-s + (0.406 − 0.913i)17-s + (0.866 − 0.5i)18-s + (0.104 + 0.994i)19-s + (−0.587 + 0.809i)22-s + (−0.743 + 0.669i)23-s + (0.5 + 0.866i)24-s + ⋯
L(s)  = 1  + (0.743 − 0.669i)2-s + (−0.994 − 0.104i)3-s + (0.104 − 0.994i)4-s + (−0.809 + 0.587i)6-s + (−0.587 − 0.809i)8-s + (0.978 + 0.207i)9-s + (−0.978 + 0.207i)11-s + (−0.207 + 0.978i)12-s + (−0.951 + 0.309i)13-s + (−0.978 − 0.207i)16-s + (0.406 − 0.913i)17-s + (0.866 − 0.5i)18-s + (0.104 + 0.994i)19-s + (−0.587 + 0.809i)22-s + (−0.743 + 0.669i)23-s + (0.5 + 0.866i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.249 + 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.249 + 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(175\)    =    \(5^{2} \cdot 7\)
Sign: $0.249 + 0.968i$
Analytic conductor: \(18.8063\)
Root analytic conductor: \(18.8063\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{175} (123, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 175,\ (1:\ ),\ 0.249 + 0.968i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3824921962 + 0.2965249593i\)
\(L(\frac12)\) \(\approx\) \(0.3824921962 + 0.2965249593i\)
\(L(1)\) \(\approx\) \(0.8148173075 - 0.3034378969i\)
\(L(1)\) \(\approx\) \(0.8148173075 - 0.3034378969i\)

Euler product

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

−26.76167329221153691204022317188, −26.11989184647240360351576251522, −24.71062072566973677797994006560, −23.98684512492522835468232620830, −23.25402736834497673792153185034, −22.248835654485220680763406317413, −21.605406765284850452758122559236, −20.63150599908359330676625512273, −19.05188237234864760777756852914, −17.693679740539057814178739416575, −17.205603145049793623496724254547, −15.98560816506624134057237119325, −15.41303824162336111832343713321, −14.1349593473408983416243309454, −12.88207410996594399207511418762, −12.26160172960746189160368993877, −11.06783805909369295603835977964, −9.96712989566504464534512621264, −8.238417846595330380830947919633, −7.195323441015071550922616297596, −6.064406460336739223905800200145, −5.17451645531097049408967369022, −4.196923176192611954987545449126, −2.56654723091799678740058903517, −0.15206410877561361098763024362, 1.44885980235542374195880092165, 2.92030686147940180908496535429, 4.564404943218482058256561485422, 5.28174735210286912167352398559, 6.43212913828508302298960896922, 7.66248334842207687787585602444, 9.783918513430249294820218520212, 10.331385623817175456384255458031, 11.65529980204259077893768188701, 12.208926334693206994933210700885, 13.25442572619440617111541079437, 14.33726668365732695258550007581, 15.6012097358874899010949298919, 16.45218361700053386863115701847, 17.87453994591942239085402731952, 18.62269633364022582790366601106, 19.71352271990585404861322529285, 20.94411803671552928486326183661, 21.64491521570315396698062681807, 22.66470816626087475973097392684, 23.33102095297735861447065828849, 24.16519894538387220771172634106, 25.12894149870081653809922575138, 26.81616304518580713955272529428, 27.6684316279136158724109460055

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