Properties

Label 1-175-175.171-r1-0-0
Degree $1$
Conductor $175$
Sign $0.784 - 0.620i$
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  + (0.913 − 0.406i)2-s + (−0.669 + 0.743i)3-s + (0.669 − 0.743i)4-s + (−0.309 + 0.951i)6-s + (0.309 − 0.951i)8-s + (−0.104 − 0.994i)9-s + (−0.104 + 0.994i)11-s + (0.104 + 0.994i)12-s + (0.809 − 0.587i)13-s + (−0.104 − 0.994i)16-s + (0.978 − 0.207i)17-s + (−0.5 − 0.866i)18-s + (−0.669 − 0.743i)19-s + (0.309 + 0.951i)22-s + (0.913 − 0.406i)23-s + (0.5 + 0.866i)24-s + ⋯
L(s)  = 1  + (0.913 − 0.406i)2-s + (−0.669 + 0.743i)3-s + (0.669 − 0.743i)4-s + (−0.309 + 0.951i)6-s + (0.309 − 0.951i)8-s + (−0.104 − 0.994i)9-s + (−0.104 + 0.994i)11-s + (0.104 + 0.994i)12-s + (0.809 − 0.587i)13-s + (−0.104 − 0.994i)16-s + (0.978 − 0.207i)17-s + (−0.5 − 0.866i)18-s + (−0.669 − 0.743i)19-s + (0.309 + 0.951i)22-s + (0.913 − 0.406i)23-s + (0.5 + 0.866i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.784 - 0.620i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.784 - 0.620i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(175\)    =    \(5^{2} \cdot 7\)
Sign: $0.784 - 0.620i$
Analytic conductor: \(18.8063\)
Root analytic conductor: \(18.8063\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{175} (171, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 175,\ (1:\ ),\ 0.784 - 0.620i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.551971382 - 0.8871359930i\)
\(L(\frac12)\) \(\approx\) \(2.551971382 - 0.8871359930i\)
\(L(1)\) \(\approx\) \(1.601427102 - 0.2657117317i\)
\(L(1)\) \(\approx\) \(1.601427102 - 0.2657117317i\)

Euler product

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

−27.307563577012159353827725684069, −26.00639938402890859157232066189, −25.087574881672876624563275154042, −24.27649418672786039053771529766, −23.33985072567575238216612205639, −22.90666363384799735109972374982, −21.55702538348228534353792145315, −21.000281292486984774066193110646, −19.353676518439463045069675773086, −18.58585937558804551692065346628, −17.14477392992520330839753879064, −16.58871600422017991447468709460, −15.51787857291095975968930123746, −14.100188104885244349478662957097, −13.4670386142415420147507458709, −12.39715278885351641895623286956, −11.53198265121771513945474422059, −10.61518745146649042519338939617, −8.51531834552335165309768247583, −7.57735151282088399409932616852, −6.28610492370853586127945504163, −5.76060150952013770225359623505, −4.34999811904960416367049072823, −2.92208641795977711186632001590, −1.29182893788778573608450365316, 0.92689914041571008150878879549, 2.79601207664811698984345343791, 4.06243937263801347826486451946, 5.01290439184046603576264934370, 6.00012238139429871884446616649, 7.17875589829745399723069942392, 9.12652179763594763279995940319, 10.33816259238160918441346976231, 10.94584117926761759618878633481, 12.16669260851920652778229825012, 12.88949509209607412483418897712, 14.32033569658899597730284574956, 15.25700636339313448651251345115, 15.96276041822831108723320686702, 17.17219484662330688504485582184, 18.30620719606368721996381523792, 19.64467979711926847625932292694, 20.798963356700392569458871832534, 21.18062608131910963626031648100, 22.46802431898612214209102685569, 23.01310436586708172439953835562, 23.760111527901897769756059289042, 25.12855414491534718511063658936, 26.03197593064598557028442999926, 27.595429763813213490415539825078

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