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Properties

Label 1-175-175.83-r0-0-0
Degree $1$
Conductor $175$
Sign $-0.968 + 0.248i$
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  + (0.587 + 0.809i)2-s + (−0.951 − 0.309i)3-s + (−0.309 + 0.951i)4-s + (−0.309 − 0.951i)6-s + (−0.951 + 0.309i)8-s + (0.809 + 0.587i)9-s + (−0.809 + 0.587i)11-s + (0.587 − 0.809i)12-s + (−0.587 + 0.809i)13-s + (−0.809 − 0.587i)16-s + (−0.951 + 0.309i)17-s + i·18-s + (0.309 + 0.951i)19-s + (−0.951 − 0.309i)22-s + (−0.587 − 0.809i)23-s + 24-s + ⋯
L(s)  = 1  + (0.587 + 0.809i)2-s + (−0.951 − 0.309i)3-s + (−0.309 + 0.951i)4-s + (−0.309 − 0.951i)6-s + (−0.951 + 0.309i)8-s + (0.809 + 0.587i)9-s + (−0.809 + 0.587i)11-s + (0.587 − 0.809i)12-s + (−0.587 + 0.809i)13-s + (−0.809 − 0.587i)16-s + (−0.951 + 0.309i)17-s + i·18-s + (0.309 + 0.951i)19-s + (−0.951 − 0.309i)22-s + (−0.587 − 0.809i)23-s + 24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.968 + 0.248i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.968 + 0.248i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(175\)    =    \(5^{2} \cdot 7\)
Sign: $-0.968 + 0.248i$
Analytic conductor: \(0.812696\)
Root analytic conductor: \(0.812696\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{175} (83, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 175,\ (0:\ ),\ -0.968 + 0.248i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.08477136485 + 0.6710344490i\)
\(L(\frac12)\) \(\approx\) \(0.08477136485 + 0.6710344490i\)
\(L(1)\) \(\approx\) \(0.6454144820 + 0.4985570024i\)
\(L(1)\) \(\approx\) \(0.6454144820 + 0.4985570024i\)

Euler product

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

−27.26593334597279393042113568333, −26.43123089499340428939131510617, −24.593604041312833411242715240589, −23.89421275410479292388262075833, −22.94066845198646899379469792697, −22.10503816896232570437319393383, −21.449653296790893093590165481338, −20.39372285217732093598191626594, −19.36793923823372489834998836571, −18.14387507633122586054707966847, −17.48962791740799344763494627928, −15.86284960914257831859425899602, −15.32591702739054573793987177237, −13.77825505210041061958865249319, −12.89302450375275380026194175664, −11.87380584408438439706255620754, −10.95694406356040468447907897163, −10.21818591936789310495100451279, −9.0507486434013061925467095747, −7.18302300939542769906096534484, −5.73795384253506985204142789745, −5.08700391749387491784801041367, −3.821871181525172990974351720481, −2.4124122595640504122034548594, −0.485364613186329620367917158646, 2.21068031232296435927935911146, 4.170390527460566014878737701527, 5.06365447251871176906948980846, 6.18281271511041836773313272724, 7.10648357005713286255823241680, 8.0849017288814969117519927380, 9.66838948904297265560396316810, 11.037458736707192556775252000665, 12.22199526090850785969483567217, 12.85374318623664427795403811062, 14.009663007847245641766911360485, 15.17354187837885087346271505331, 16.19993343133172058074289698935, 16.93002336404092373429624415604, 17.94395158145142340426022430976, 18.68734968302945561977194429274, 20.37398083541901551512737998770, 21.555586166036199739312061557563, 22.33796725990539454142810113111, 23.1480413455025342899454462875, 24.07326228727313998682381747063, 24.602464762542667623165790123337, 25.92101762466914044309792273205, 26.72751246162164942451393198211, 27.88023888018711241353936646522

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