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Properties

Label 1-175-175.102-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: $\chi_{175} (102, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 175,\ (1:\ ),\ 0.249 + 0.968i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.424863940 + 1.104617888i\)
\(L(\frac12)\) \(\approx\) \(1.424863940 + 1.104617888i\)
\(L(1)\) \(\approx\) \(1.039919042 + 0.4191162953i\)
\(L(1)\) \(\approx\) \(1.039919042 + 0.4191162953i\)

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

−26.82950679032103012661400244970, −26.21250693112803634610200524923, −25.4140006298902109173372518664, −24.42548972916433593948970621366, −23.14008933056138213410803543281, −21.66594168247253872315229928121, −20.95578721937188006964159932868, −20.21470512919765190507959103665, −19.18063204523029291851401126839, −18.48102394158045225666385115706, −17.56548125509417631374974144953, −16.0492339593801431003734963362, −15.422313642778434308579660064664, −13.58285108358444270071288542914, −13.30089156480755639140576483943, −11.80656019196610481778475418805, −10.74330752073179594524372200568, −9.60639925995753838154900072059, −8.74110863065158021802882948714, −7.83926986579386359958551994708, −6.765903472800845172416514750409, −4.61592461718524141329017329989, −3.23105329839481342837299734483, −2.366040484521574299790450947412, −0.84783190994157487320836398120, 1.28284549706700506789498773417, 2.70318887940132824081970796809, 4.34094407512916698376879395987, 5.798456228299640041025325805820, 7.095809654919173460794801225685, 8.20102038544576885511986393724, 8.748498832140015101840925333760, 10.12186791565540257284997606286, 10.74170653505931274285772335737, 12.71661453218577902559123227589, 13.75794303754985150485570328933, 14.78079804456907602291872534816, 15.60113842418734124464407440720, 16.3884765851554001990188719508, 17.81425662737599091076289128640, 18.59463095903423111127386212401, 19.47829985774924906557864366019, 20.48488069497042435772928195206, 21.2546310397184989265668678424, 22.92436484997833863080864700639, 23.80539062568523703476345760769, 24.88080479243974611833768024692, 25.54796859008055307327355622399, 26.39421673957811267529416293354, 27.0610464642226155319560964613

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