Properties

Label 1-175-175.34-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: $\chi_{175} (34, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 175,\ (1:\ ),\ -0.187 + 0.982i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2633672097 + 0.3183562684i\)
\(L(\frac12)\) \(\approx\) \(0.2633672097 + 0.3183562684i\)
\(L(1)\) \(\approx\) \(0.5824586302 - 0.07358163672i\)
\(L(1)\) \(\approx\) \(0.5824586302 - 0.07358163672i\)

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

−26.77440812732645011362259820485, −26.03716002903915602182322517117, −24.65283505500680131191047637648, −24.22724081403906481166050816688, −23.37878545662764105943615981716, −22.33695163676947493414696257329, −21.572004940286450461740286605448, −19.62675752939924302618582809025, −18.90340626061705062340858250780, −17.96169473459338355381002737924, −17.0899598156237416030912673575, −16.306143524532776788783004906590, −15.359259917325772563080609295225, −13.81949049647821090241713459241, −13.36979338884881695638529666968, −11.77286130596428868111533194495, −10.87727980895107831165347961597, −9.460726502424340102111721141698, −8.34804936850924506513205675746, −7.179551486133372981706602848062, −6.291751811431024074458170512645, −5.40477644401512397253507605180, −4.049784880111157905833759738346, −1.632254133193134758000219341350, −0.20666164734757702458229909577, 1.30282980084778761275041082378, 3.05433055834011106768682364467, 4.319468661144668659062883328036, 5.26026729686465745434857705138, 6.85509176433739185941978823273, 8.36717299525120114081330225810, 9.6501114124724011079290358605, 10.29082340675444936397989852022, 11.37901772605068381907806834311, 12.19951555822167555923580742113, 13.16535076938226390196431283061, 14.626463772274298496061502045215, 15.84763104379424290387309026638, 16.9214422120256411204923982342, 17.8796036967544705510590739158, 18.43823341011719794495838974731, 20.15092049292703614034786202986, 20.4776137696191380322252059245, 21.71421365786638479807195185056, 22.65479665659713350774063748929, 22.97805903137641475485755245416, 24.59077778633803591820297896793, 25.908662785898974762543890964328, 26.895284039965064672659021185860, 27.62547821030372105436635377502

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