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Properties

Label 1-175-175.41-r1-0-0
Degree $1$
Conductor $175$
Sign $-0.187 + 0.982i$
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  + (0.309 + 0.951i)2-s + (0.809 − 0.587i)3-s + (−0.809 + 0.587i)4-s + (0.809 + 0.587i)6-s + (−0.809 − 0.587i)8-s + (0.309 − 0.951i)9-s + (0.309 + 0.951i)11-s + (−0.309 + 0.951i)12-s + (−0.309 + 0.951i)13-s + (0.309 − 0.951i)16-s + (0.809 + 0.587i)17-s + 18-s + (0.809 + 0.587i)19-s + (−0.809 + 0.587i)22-s + (0.309 + 0.951i)23-s − 24-s + ⋯
L(s)  = 1  + (0.309 + 0.951i)2-s + (0.809 − 0.587i)3-s + (−0.809 + 0.587i)4-s + (0.809 + 0.587i)6-s + (−0.809 − 0.587i)8-s + (0.309 − 0.951i)9-s + (0.309 + 0.951i)11-s + (−0.309 + 0.951i)12-s + (−0.309 + 0.951i)13-s + (0.309 − 0.951i)16-s + (0.809 + 0.587i)17-s + 18-s + (0.809 + 0.587i)19-s + (−0.809 + 0.587i)22-s + (0.309 + 0.951i)23-s − 24-s + ⋯

Functional equation

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

Invariants

Degree: \(1\)
Conductor: \(175\)    =    \(5^{2} \cdot 7\)
Sign: $-0.187 + 0.982i$
Analytic conductor: \(18.8063\)
Root analytic conductor: \(18.8063\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{175} (41, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 175,\ (1:\ ),\ -0.187 + 0.982i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.602089059 + 1.936592999i\)
\(L(\frac12)\) \(\approx\) \(1.602089059 + 1.936592999i\)
\(L(1)\) \(\approx\) \(1.346897671 + 0.7404632611i\)
\(L(1)\) \(\approx\) \(1.346897671 + 0.7404632611i\)

Euler product

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

−27.11655373376027367292276496267, −26.28874916862928662193317190529, −24.94794174513385487721234320106, −24.09796480005762830593870124295, −22.533035573320691407299143170442, −22.131989061214760055583253504311, −20.84508039157502574551458044409, −20.43857986091051287007283316687, −19.28432095251223710919477877297, −18.64894823046486052334730525943, −17.194600627372885837934226985921, −15.86296491658355396277795751084, −14.796055030720677512743963608583, −13.93087142390475068083325952221, −13.11401863517541106042467992408, −11.80095600284089479288774133186, −10.72343613667448089665132693794, −9.79203980439526477996097501735, −8.87804618071080931476467168272, −7.761179371111080771872526051484, −5.728556704258640615366245489146, −4.62844510664744113137853756091, −3.35830401487185766401836487795, −2.61726850869727993663871742812, −0.809959325526999738523323143827, 1.52877832828732498212068541600, 3.254818632708645098963360496085, 4.379092029548370282823960721333, 5.872739528666251849750505277852, 7.116217482433979308305264439291, 7.69763991575623719577489338073, 9.00657469165401673587637060312, 9.76317430248006867605579509254, 11.9608691446028487675789773219, 12.68805976827419439035468214134, 13.87591692504398132343675926498, 14.53090478570019669290857039898, 15.400670866406966450751747717528, 16.64339380930887955459480446256, 17.65045582632293681254134327567, 18.59914828672202714931027178553, 19.55931148990994875465554263572, 20.79529894077682330094541890697, 21.73908955959078674421217728895, 23.0402678507247174505888651926, 23.74087788778849498356312043630, 24.73833020153043346275965945999, 25.42110690623842231966827876504, 26.22521412116776215346913028661, 27.06893430756928040230901383191

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