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Properties

Label 1-235-235.123-r0-0-0
Degree $1$
Conductor $235$
Sign $0.463 + 0.886i$
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  + (−0.942 + 0.334i)2-s + (−0.979 + 0.203i)3-s + (0.775 − 0.631i)4-s + (0.854 − 0.519i)6-s + (0.816 + 0.576i)7-s + (−0.519 + 0.854i)8-s + (0.917 − 0.398i)9-s + (−0.460 + 0.887i)11-s + (−0.631 + 0.775i)12-s + (−0.730 − 0.682i)13-s + (−0.962 − 0.269i)14-s + (0.203 − 0.979i)16-s + (0.887 − 0.460i)17-s + (−0.730 + 0.682i)18-s + (−0.0682 − 0.997i)19-s + ⋯
L(s)  = 1  + (−0.942 + 0.334i)2-s + (−0.979 + 0.203i)3-s + (0.775 − 0.631i)4-s + (0.854 − 0.519i)6-s + (0.816 + 0.576i)7-s + (−0.519 + 0.854i)8-s + (0.917 − 0.398i)9-s + (−0.460 + 0.887i)11-s + (−0.631 + 0.775i)12-s + (−0.730 − 0.682i)13-s + (−0.962 − 0.269i)14-s + (0.203 − 0.979i)16-s + (0.887 − 0.460i)17-s + (−0.730 + 0.682i)18-s + (−0.0682 − 0.997i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.463 + 0.886i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.463 + 0.886i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(235\)    =    \(5 \cdot 47\)
Sign: $0.463 + 0.886i$
Analytic conductor: \(1.09133\)
Root analytic conductor: \(1.09133\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{235} (123, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 235,\ (0:\ ),\ 0.463 + 0.886i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5052808607 + 0.3059598761i\)
\(L(\frac12)\) \(\approx\) \(0.5052808607 + 0.3059598761i\)
\(L(1)\) \(\approx\) \(0.5641086318 + 0.1687053326i\)
\(L(1)\) \(\approx\) \(0.5641086318 + 0.1687053326i\)

Euler product

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

−26.54435903649874839932457074293, −25.169102192267666384190668396963, −24.19602147336753469312825445762, −23.60511166652809011846178593444, −22.25644342163386751356621722268, −21.23657645527739217428838610992, −20.71442153530039945477252811202, −19.0548222216673502447182206146, −18.80407950250036186632111022292, −17.52575804667551999674189395899, −16.90657632388682266124590601389, −16.28488479054863667170085284242, −14.89761808915082015059678849831, −13.50485313312434877978249875846, −12.25685383921032930100969020689, −11.53329936469120575413446783088, −10.62732912931592143382025547127, −9.92269852687937287440669131553, −8.35123213517555848928489190324, −7.55001367174188146120986974346, −6.48473198326051075890487934399, −5.22469754074564010975384489337, −3.806864974106436157364147624280, −2.04182335276727311829784588673, −0.81141774743590486864765580650, 1.147896270835391706689609752865, 2.629585680122703646518878004049, 4.98277015090294643535757259358, 5.369116932156613557528515188395, 6.900775355519948488424635626826, 7.632135629370227447609177806113, 8.97883409627303463110471728468, 10.01275095345073632885998153799, 10.837715368085660369931394060344, 11.80755375430470039160499645569, 12.653935039904713010379120213, 14.57759484083051458105120625937, 15.326404909087835919483988195046, 16.14629676716564539140574434957, 17.36158831567626147726756396582, 17.76872635345123988161567738355, 18.566612324935614794827876928518, 19.78336903193265755484165194092, 20.88736448975119284257619295525, 21.67276891542172628318585718722, 23.00360497630812684694675295167, 23.70367192136416407680903667405, 24.72296729925192424611000101994, 25.426205580725199275275648952493, 26.71547210953729396939530896817

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