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Properties

Label 1-235-235.62-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: $\chi_{235} (62, \cdot )$
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

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

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