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Properties

Label 1-175-175.108-r0-0-0
Degree $1$
Conductor $175$
Sign $-0.388 + 0.921i$
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  + (−0.994 + 0.104i)2-s + (0.207 + 0.978i)3-s + (0.978 − 0.207i)4-s + (−0.309 − 0.951i)6-s + (−0.951 + 0.309i)8-s + (−0.913 + 0.406i)9-s + (0.913 + 0.406i)11-s + (0.406 + 0.913i)12-s + (−0.587 + 0.809i)13-s + (0.913 − 0.406i)16-s + (0.743 + 0.669i)17-s + (0.866 − 0.5i)18-s + (−0.978 − 0.207i)19-s + (−0.951 − 0.309i)22-s + (0.994 − 0.104i)23-s + (−0.5 − 0.866i)24-s + ⋯
L(s)  = 1  + (−0.994 + 0.104i)2-s + (0.207 + 0.978i)3-s + (0.978 − 0.207i)4-s + (−0.309 − 0.951i)6-s + (−0.951 + 0.309i)8-s + (−0.913 + 0.406i)9-s + (0.913 + 0.406i)11-s + (0.406 + 0.913i)12-s + (−0.587 + 0.809i)13-s + (0.913 − 0.406i)16-s + (0.743 + 0.669i)17-s + (0.866 − 0.5i)18-s + (−0.978 − 0.207i)19-s + (−0.951 − 0.309i)22-s + (0.994 − 0.104i)23-s + (−0.5 − 0.866i)24-s + ⋯

Functional equation

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

Invariants

Degree: \(1\)
Conductor: \(175\)    =    \(5^{2} \cdot 7\)
Sign: $-0.388 + 0.921i$
Analytic conductor: \(0.812696\)
Root analytic conductor: \(0.812696\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{175} (108, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 175,\ (0:\ ),\ -0.388 + 0.921i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3940082092 + 0.5937296675i\)
\(L(\frac12)\) \(\approx\) \(0.3940082092 + 0.5937296675i\)
\(L(1)\) \(\approx\) \(0.6294529673 + 0.3588893784i\)
\(L(1)\) \(\approx\) \(0.6294529673 + 0.3588893784i\)

Euler product

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

−27.26966388324886915751842229151, −26.112679145796526551951125876538, −25.086011298775114728793972864779, −24.74476530388724908607710913854, −23.55996190055648598967761164434, −22.3706042844216173967001504637, −20.950539144261289482015457762904, −20.0606863072051927595551822582, −19.133415530415707833542092431093, −18.59854005353724122369062539698, −17.27129760847327277819354380769, −16.919412168417350395962693664098, −15.30503424860591527711797962499, −14.34381557185425024075152857136, −12.94164122458438645348627689733, −12.01629055033500984988221479644, −11.09949073297506179853149118704, −9.72330036906100992808620423560, −8.69347091884404713127416858833, −7.72736000710837559554840400826, −6.81240225857591225249199493808, −5.70428313457718892931671976402, −3.3817831192978423193147846299, −2.18022532237159137344146411312, −0.7871360839560617062495109776, 1.79942757623693140275954096327, 3.3009200764202826759925299940, 4.70446571955487654927560738103, 6.18579651274628109756205690780, 7.38556664481690051923460653696, 8.77650618824497635087109471777, 9.35660880533420410602123157330, 10.43659353881100420886098005114, 11.3113839653133407888507519617, 12.496578846303417640409744973650, 14.59925155840471902879996432207, 14.798071051913827510892036898751, 16.29221206203720210060727188687, 16.822252817625811943009134441315, 17.76998142589244302575518359593, 19.33985726445888714615162660963, 19.66338484295553636391420741128, 21.0094959005024369683794800471, 21.57518812687135395193453380135, 22.90368551774080790531472926911, 24.13367523172811137413702084573, 25.31176325374513962874531250226, 25.90816823828744851155320343806, 26.90826647056090337161949658993, 27.62049202065825880151670404365

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