Properties

Label 1-175-175.31-r1-0-0
Degree $1$
Conductor $175$
Sign $0.388 - 0.921i$
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  + (−0.104 − 0.994i)2-s + (0.978 − 0.207i)3-s + (−0.978 + 0.207i)4-s + (−0.309 − 0.951i)6-s + (0.309 + 0.951i)8-s + (0.913 − 0.406i)9-s + (0.913 + 0.406i)11-s + (−0.913 + 0.406i)12-s + (0.809 + 0.587i)13-s + (0.913 − 0.406i)16-s + (−0.669 + 0.743i)17-s + (−0.5 − 0.866i)18-s + (0.978 + 0.207i)19-s + (0.309 − 0.951i)22-s + (−0.104 − 0.994i)23-s + (0.5 + 0.866i)24-s + ⋯
L(s)  = 1  + (−0.104 − 0.994i)2-s + (0.978 − 0.207i)3-s + (−0.978 + 0.207i)4-s + (−0.309 − 0.951i)6-s + (0.309 + 0.951i)8-s + (0.913 − 0.406i)9-s + (0.913 + 0.406i)11-s + (−0.913 + 0.406i)12-s + (0.809 + 0.587i)13-s + (0.913 − 0.406i)16-s + (−0.669 + 0.743i)17-s + (−0.5 − 0.866i)18-s + (0.978 + 0.207i)19-s + (0.309 − 0.951i)22-s + (−0.104 − 0.994i)23-s + (0.5 + 0.866i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.388 - 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.388 - 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(175\)    =    \(5^{2} \cdot 7\)
Sign: $0.388 - 0.921i$
Analytic conductor: \(18.8063\)
Root analytic conductor: \(18.8063\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{175} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 175,\ (1:\ ),\ 0.388 - 0.921i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.088077547 - 1.385680622i\)
\(L(\frac12)\) \(\approx\) \(2.088077547 - 1.385680622i\)
\(L(1)\) \(\approx\) \(1.318211798 - 0.6619309654i\)
\(L(1)\) \(\approx\) \(1.318211798 - 0.6619309654i\)

Euler product

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

−27.27673473365727790498473355410, −26.237909059467045876645619055025, −25.44765550371485951211991851370, −24.71421923975771373063053119131, −23.88869152453368617974151235014, −22.54578098189364799306546243576, −21.82272429371680699502453090789, −20.45616066473163465276259101111, −19.56658210883054747578010882130, −18.49945824670580037933534085539, −17.61353379178178843930698798765, −16.20047227050598516435313679859, −15.66524510133999934249914496893, −14.5137315026229346781310118290, −13.795361886966282513978568925837, −12.92233781934882366778052997949, −11.15072232959434069666322362086, −9.62320734587818825575714524098, −9.00626241597053642722165355659, −7.934875189673221584967938283369, −6.96842972220713291162529849495, −5.6417192632857760528214417511, −4.26750199230310668733048411741, −3.20011784895981737341857006507, −1.14500430899870524195858894052, 1.18926287078292399506142971652, 2.25484360610908650247394042391, 3.61779357237470876158175391650, 4.426955566539453446606741590008, 6.42029151218231651695533628146, 7.89302516442693423151135486601, 8.91746426169031523752460737418, 9.62676888674204851163512557733, 10.90244177664270689842522084791, 12.07300052736263176575330158767, 13.01952669875251294519533084820, 13.99792005115591100748940990245, 14.74209905146773225592718290574, 16.21664447835064314372705626779, 17.64356239102102782566423835637, 18.499822596315368777550859666, 19.46947966473504411257764855120, 20.13639732925527965679880676512, 21.014549338255948697716672139263, 21.93158132546847748767895257688, 23.003119386139801418098762903746, 24.188163352156597469473039295492, 25.25095506059074454448223620472, 26.3293949666364328138185267121, 26.91478634331051704354296564886

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