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Properties

Label 1-175-175.48-r0-0-0
Degree $1$
Conductor $175$
Sign $0.982 - 0.187i$
Analytic cond. $0.812696$
Root an. cond. $0.812696$
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  + (−0.951 − 0.309i)2-s + (−0.587 + 0.809i)3-s + (0.809 + 0.587i)4-s + (0.809 − 0.587i)6-s + (−0.587 − 0.809i)8-s + (−0.309 − 0.951i)9-s + (0.309 − 0.951i)11-s + (−0.951 + 0.309i)12-s + (0.951 − 0.309i)13-s + (0.309 + 0.951i)16-s + (−0.587 − 0.809i)17-s + i·18-s + (−0.809 + 0.587i)19-s + (−0.587 + 0.809i)22-s + (0.951 + 0.309i)23-s + 24-s + ⋯
L(s)  = 1  + (−0.951 − 0.309i)2-s + (−0.587 + 0.809i)3-s + (0.809 + 0.587i)4-s + (0.809 − 0.587i)6-s + (−0.587 − 0.809i)8-s + (−0.309 − 0.951i)9-s + (0.309 − 0.951i)11-s + (−0.951 + 0.309i)12-s + (0.951 − 0.309i)13-s + (0.309 + 0.951i)16-s + (−0.587 − 0.809i)17-s + i·18-s + (−0.809 + 0.587i)19-s + (−0.587 + 0.809i)22-s + (0.951 + 0.309i)23-s + 24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.982 - 0.187i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.982 - 0.187i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(175\)    =    \(5^{2} \cdot 7\)
Sign: $0.982 - 0.187i$
Analytic conductor: \(0.812696\)
Root analytic conductor: \(0.812696\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{175} (48, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 175,\ (0:\ ),\ 0.982 - 0.187i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6103846203 - 0.05769833434i\)
\(L(\frac12)\) \(\approx\) \(0.6103846203 - 0.05769833434i\)
\(L(1)\) \(\approx\) \(0.6256848110 + 0.0001287501421i\)
\(L(1)\) \(\approx\) \(0.6256848110 + 0.0001287501421i\)

Euler product

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

−27.68101246590634841142599781729, −26.44440165186031325717665340933, −25.43193483547776811826723958753, −24.83316944211419993550285168473, −23.616925326482791428889762806733, −23.16132732839438528262760293496, −21.66435574553601415775636171551, −20.34675520368186505796330892773, −19.38235028431007179246215990978, −18.62278066619299742301664311794, −17.55047316587133274254527379091, −17.11716915030507162669713996323, −15.86842652794859736329339597576, −14.86131126998211477107585055771, −13.46247863578704876048688450796, −12.311729397148769669369439461731, −11.23699798484586023710709065801, −10.42034408073819182773261064693, −8.97499065381452928232968122824, −8.031331541737817417794918661765, −6.759691135812697113105295146, −6.26488562284638244301406372013, −4.68317826600368139478681851620, −2.38100513873547292252632682425, −1.194784454131137504681683680864, 0.902933589127519185669664247865, 2.92530532573377833583972708625, 4.08098181414438927110041410383, 5.768945018923285471107326483806, 6.75922751359596147988682046375, 8.41013957785918309733498723582, 9.14623303039400235573578877211, 10.36719753441682626415346329780, 11.10718122291347418384922752250, 11.91300858788959826256281769516, 13.32494460936096792311261061762, 14.95468001469659165455087930933, 15.983531071526156475051656760426, 16.61970798452910754010930120474, 17.61996405741941537608765430379, 18.51918031644287784454051708129, 19.61672223997207752951986595240, 20.77359976087213274361272256115, 21.34348119069375837543903458102, 22.408240590132086580710056614559, 23.49850079056494176604292771710, 24.80148911152677174860726534950, 25.7423466708150644139208970091, 26.85069594612297658132441514623, 27.325676320555698124909388733664

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