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Properties

Label 1-235-235.163-r0-0-0
Degree $1$
Conductor $235$
Sign $0.467 + 0.883i$
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  + (−0.979 − 0.203i)2-s + (0.730 + 0.682i)3-s + (0.917 + 0.398i)4-s + (−0.576 − 0.816i)6-s + (0.631 + 0.775i)7-s + (−0.816 − 0.576i)8-s + (0.0682 + 0.997i)9-s + (0.334 − 0.942i)11-s + (0.398 + 0.917i)12-s + (0.136 + 0.990i)13-s + (−0.460 − 0.887i)14-s + (0.682 + 0.730i)16-s + (−0.942 + 0.334i)17-s + (0.136 − 0.990i)18-s + (0.962 − 0.269i)19-s + ⋯
L(s)  = 1  + (−0.979 − 0.203i)2-s + (0.730 + 0.682i)3-s + (0.917 + 0.398i)4-s + (−0.576 − 0.816i)6-s + (0.631 + 0.775i)7-s + (−0.816 − 0.576i)8-s + (0.0682 + 0.997i)9-s + (0.334 − 0.942i)11-s + (0.398 + 0.917i)12-s + (0.136 + 0.990i)13-s + (−0.460 − 0.887i)14-s + (0.682 + 0.730i)16-s + (−0.942 + 0.334i)17-s + (0.136 − 0.990i)18-s + (0.962 − 0.269i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.467 + 0.883i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.467 + 0.883i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(235\)    =    \(5 \cdot 47\)
Sign: $0.467 + 0.883i$
Analytic conductor: \(1.09133\)
Root analytic conductor: \(1.09133\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{235} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 235,\ (0:\ ),\ 0.467 + 0.883i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9109463041 + 0.5485753514i\)
\(L(\frac12)\) \(\approx\) \(0.9109463041 + 0.5485753514i\)
\(L(1)\) \(\approx\) \(0.9126707556 + 0.2702412378i\)
\(L(1)\) \(\approx\) \(0.9126707556 + 0.2702412378i\)

Euler product

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

−26.10986731965599451331001495800, −25.19533063612334070436277969094, −24.58419998330220347832784476179, −23.70456977918363851787182688240, −22.65534674921278895360028106475, −20.80308504069335079935132643986, −20.30404854264280650181800695372, −19.68742779073149360031038725140, −18.45228997038979563735755405575, −17.77151845109183634973912294775, −17.09394584341980364672779573762, −15.61168187080579578896471826503, −14.85195335488119506161758280522, −13.87742772020474922491837288234, −12.69722103347407725218550575881, −11.5396263393230809854837749140, −10.45458957316920459364600876427, −9.36504999916986488981699194133, −8.448295083224670596797182149334, −7.332525442287709020631065926467, −7.00561970306532523075965651747, −5.31720949991564406817443636204, −3.53457862482183289145784583139, −2.1235327467528496258734937916, −1.06574133709649429827391818457, 1.70012405867804326736777369417, 2.77585072593347558725442620202, 4.00230991027414273474401828828, 5.53268823543418598799248317030, 6.99279802238909292280429954292, 8.262642115241213942742769693, 8.93445983425954493595065441967, 9.592996189109899458244547714135, 11.15687293956368245793106974943, 11.38167732580577263090834620288, 13.06004505712229898420906033590, 14.3421864378527935178369716019, 15.2328285025732715556258633779, 16.13669474569630980496229123565, 16.93921473114292970047311880070, 18.21327115159491405902769181797, 19.01301026385026721872889828196, 19.80103793574045632481817446938, 20.86543555922747723319960107551, 21.476610317679170074116807126297, 22.25087740205995083243538063540, 24.420561749872071839738131265332, 24.50966530057036276225718469660, 25.8272503431199552291071102815, 26.55063158633571969565669376421

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