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Properties

Label 1-232-232.107-r1-0-0
Degree $1$
Conductor $232$
Sign $0.995 - 0.0954i$
Analytic cond. $24.9318$
Root an. cond. $24.9318$
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  + (−0.222 + 0.974i)3-s + (−0.623 + 0.781i)5-s + (0.222 − 0.974i)7-s + (−0.900 − 0.433i)9-s + (−0.900 + 0.433i)11-s + (0.900 − 0.433i)13-s + (−0.623 − 0.781i)15-s + 17-s + (−0.222 − 0.974i)19-s + (0.900 + 0.433i)21-s + (−0.623 − 0.781i)23-s + (−0.222 − 0.974i)25-s + (0.623 − 0.781i)27-s + (−0.623 + 0.781i)31-s + (−0.222 − 0.974i)33-s + ⋯
L(s)  = 1  + (−0.222 + 0.974i)3-s + (−0.623 + 0.781i)5-s + (0.222 − 0.974i)7-s + (−0.900 − 0.433i)9-s + (−0.900 + 0.433i)11-s + (0.900 − 0.433i)13-s + (−0.623 − 0.781i)15-s + 17-s + (−0.222 − 0.974i)19-s + (0.900 + 0.433i)21-s + (−0.623 − 0.781i)23-s + (−0.222 − 0.974i)25-s + (0.623 − 0.781i)27-s + (−0.623 + 0.781i)31-s + (−0.222 − 0.974i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 232 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.995 - 0.0954i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 232 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.995 - 0.0954i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(232\)    =    \(2^{3} \cdot 29\)
Sign: $0.995 - 0.0954i$
Analytic conductor: \(24.9318\)
Root analytic conductor: \(24.9318\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{232} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 232,\ (1:\ ),\ 0.995 - 0.0954i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.224459568 - 0.05854050931i\)
\(L(\frac12)\) \(\approx\) \(1.224459568 - 0.05854050931i\)
\(L(1)\) \(\approx\) \(0.8685011845 + 0.1765864172i\)
\(L(1)\) \(\approx\) \(0.8685011845 + 0.1765864172i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
29 \( 1 \)
good3 \( 1 + (-0.222 + 0.974i)T \)
5 \( 1 + (-0.623 + 0.781i)T \)
7 \( 1 + (0.222 - 0.974i)T \)
11 \( 1 + (-0.900 + 0.433i)T \)
13 \( 1 + (0.900 - 0.433i)T \)
17 \( 1 + T \)
19 \( 1 + (-0.222 - 0.974i)T \)
23 \( 1 + (-0.623 - 0.781i)T \)
31 \( 1 + (-0.623 + 0.781i)T \)
37 \( 1 + (0.900 + 0.433i)T \)
41 \( 1 + T \)
43 \( 1 + (0.623 + 0.781i)T \)
47 \( 1 + (0.900 - 0.433i)T \)
53 \( 1 + (-0.623 + 0.781i)T \)
59 \( 1 + T \)
61 \( 1 + (0.222 - 0.974i)T \)
67 \( 1 + (-0.900 - 0.433i)T \)
71 \( 1 + (0.900 - 0.433i)T \)
73 \( 1 + (0.623 + 0.781i)T \)
79 \( 1 + (0.900 + 0.433i)T \)
83 \( 1 + (-0.222 - 0.974i)T \)
89 \( 1 + (0.623 - 0.781i)T \)
97 \( 1 + (-0.222 - 0.974i)T \)
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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

−25.73551708108216345184654377879, −25.165923659362455276023142922647, −23.95940147745338921907464113351, −23.72123459546200316991002954205, −22.64232128681078125587428787614, −21.27955783693609584264106720650, −20.579738266787943103762618671026, −19.23062041453713102578005405178, −18.7245083509625285259278601009, −17.85409668602659858707023730676, −16.5435355780466209800418587870, −15.9219352253367725860481799559, −14.62305402946618565804917795883, −13.45177305759221024140887337815, −12.561600480462718168267572177652, −11.85721560127032551197334299379, −10.96459132854586366107404222398, −9.23538268501403710443597115527, −8.176603713457898496296481352244, −7.69005717084958468062474277169, −5.94115875213620915945406806101, −5.42952920253602301958645976774, −3.750832591145921906934696040963, −2.22608318509916962225755716474, −0.97655647093968507548250904388, 0.52869661623706453123003026286, 2.81990573285786370558470514489, 3.82962221507964088895781342940, 4.76946191291867620191976849608, 6.11467252036064114539372388418, 7.40497739363063236955736419021, 8.32111800500825956313030035374, 9.84415823617641658051163435579, 10.70322383983208003721113431844, 11.14117102258320702481717184355, 12.548691861195503908929605889296, 13.92737045458996637967689640783, 14.784595524240823811073016567568, 15.70450802289820370294501728682, 16.41091925360090419794340501569, 17.62425392835808314430341093295, 18.42887261318039378843125006991, 19.79883531460212276186474187334, 20.51193443037838906623321221932, 21.393556095721279040997755751372, 22.500078238705093811326050832541, 23.28543577718695873762048813679, 23.725072952049166548249151420049, 25.62630607329919474417504539248, 26.18337911120923393658556850414

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