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Properties

Label 1-235-235.49-r0-0-0
Degree $1$
Conductor $235$
Sign $0.746 - 0.665i$
Analytic cond. $1.09133$
Root an. cond. $1.09133$
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  + (−0.962 − 0.269i)2-s + (−0.460 + 0.887i)3-s + (0.854 + 0.519i)4-s + (0.682 − 0.730i)6-s + (0.990 + 0.136i)7-s + (−0.682 − 0.730i)8-s + (−0.576 − 0.816i)9-s + (−0.0682 − 0.997i)11-s + (−0.854 + 0.519i)12-s + (0.334 − 0.942i)13-s + (−0.917 − 0.398i)14-s + (0.460 + 0.887i)16-s + (0.0682 − 0.997i)17-s + (0.334 + 0.942i)18-s + (−0.775 − 0.631i)19-s + ⋯
L(s)  = 1  + (−0.962 − 0.269i)2-s + (−0.460 + 0.887i)3-s + (0.854 + 0.519i)4-s + (0.682 − 0.730i)6-s + (0.990 + 0.136i)7-s + (−0.682 − 0.730i)8-s + (−0.576 − 0.816i)9-s + (−0.0682 − 0.997i)11-s + (−0.854 + 0.519i)12-s + (0.334 − 0.942i)13-s + (−0.917 − 0.398i)14-s + (0.460 + 0.887i)16-s + (0.0682 − 0.997i)17-s + (0.334 + 0.942i)18-s + (−0.775 − 0.631i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.746 - 0.665i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.746 - 0.665i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(235\)    =    \(5 \cdot 47\)
Sign: $0.746 - 0.665i$
Analytic conductor: \(1.09133\)
Root analytic conductor: \(1.09133\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{235} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 235,\ (0:\ ),\ 0.746 - 0.665i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6020099980 - 0.2293651369i\)
\(L(\frac12)\) \(\approx\) \(0.6020099980 - 0.2293651369i\)
\(L(1)\) \(\approx\) \(0.6459133800 - 0.04252674078i\)
\(L(1)\) \(\approx\) \(0.6459133800 - 0.04252674078i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
47 \( 1 \)
good2 \( 1 + (-0.962 - 0.269i)T \)
3 \( 1 + (-0.460 + 0.887i)T \)
7 \( 1 + (0.990 + 0.136i)T \)
11 \( 1 + (-0.0682 - 0.997i)T \)
13 \( 1 + (0.334 - 0.942i)T \)
17 \( 1 + (0.0682 - 0.997i)T \)
19 \( 1 + (-0.775 - 0.631i)T \)
23 \( 1 + (-0.962 + 0.269i)T \)
29 \( 1 + (-0.334 - 0.942i)T \)
31 \( 1 + (0.460 + 0.887i)T \)
37 \( 1 + (0.917 - 0.398i)T \)
41 \( 1 + (0.682 - 0.730i)T \)
43 \( 1 + (-0.854 - 0.519i)T \)
53 \( 1 + (-0.682 + 0.730i)T \)
59 \( 1 + (0.854 - 0.519i)T \)
61 \( 1 + (-0.917 - 0.398i)T \)
67 \( 1 + (0.990 - 0.136i)T \)
71 \( 1 + (0.962 - 0.269i)T \)
73 \( 1 + (0.576 - 0.816i)T \)
79 \( 1 + (0.203 + 0.979i)T \)
83 \( 1 + (0.0682 + 0.997i)T \)
89 \( 1 + (-0.775 + 0.631i)T \)
97 \( 1 + (-0.460 + 0.887i)T \)
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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

−26.200136876100380267645734210108, −25.539912217311641770483746888897, −24.48876655800100010886068124484, −23.82724021693506819833379280776, −23.16263736884952148014370278239, −21.60786450403794979190693291227, −20.53009053773151165956505165075, −19.62937661322085780190327556657, −18.6029920526008540363929342980, −18.0025745208094615834276855921, −17.13119589814447968283478862450, −16.4585833833524953988420038789, −14.97120400054390085746217553759, −14.26357864474606985380819171557, −12.77891014982650804120281086363, −11.75242985373048531639649256490, −10.98441836461935700940261616245, −9.925349694236122743967638419829, −8.44802900388197216561711018972, −7.8356796223480246119790745230, −6.75677018552831348875128531564, −5.88833357000475261200575834501, −4.472269846014481860208895660896, −2.11402071833038554680821430339, −1.48232900045299233627906546853, 0.70866317163818928086234678369, 2.54316800373894042276317814810, 3.79842731747353989876660795811, 5.233482753189726290386726276747, 6.29416697296209289475616255312, 7.8707070249645512542055032092, 8.66275772359202938891553233725, 9.70091299089005710009819342442, 10.83167009273306313299922492851, 11.258518226724755319487681419189, 12.289133595394833146594934272800, 13.89914356614774670362822948066, 15.24208405313987281539141247174, 15.874875481403350584307609401141, 16.89856189412946944867750088762, 17.71941965073645775898451869678, 18.41944641752990475257620364692, 19.744483114767568344960253182992, 20.66660126766841474180818596552, 21.33177617382585723702765017330, 22.11584508762552589687942944728, 23.43825325655675439527849713705, 24.48838259804505814050328395998, 25.43770698848662450127321017474, 26.56866346136341971046341420873

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