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Properties

Label 1-235-235.77-r0-0-0
Degree $1$
Conductor $235$
Sign $0.306 - 0.951i$
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  + (−0.997 − 0.0682i)2-s + (−0.269 − 0.962i)3-s + (0.990 + 0.136i)4-s + (0.203 + 0.979i)6-s + (−0.730 + 0.682i)7-s + (−0.979 − 0.203i)8-s + (−0.854 + 0.519i)9-s + (0.917 − 0.398i)11-s + (−0.136 − 0.990i)12-s + (0.887 + 0.460i)13-s + (0.775 − 0.631i)14-s + (0.962 + 0.269i)16-s + (0.398 − 0.917i)17-s + (0.887 − 0.460i)18-s + (−0.576 − 0.816i)19-s + ⋯
L(s)  = 1  + (−0.997 − 0.0682i)2-s + (−0.269 − 0.962i)3-s + (0.990 + 0.136i)4-s + (0.203 + 0.979i)6-s + (−0.730 + 0.682i)7-s + (−0.979 − 0.203i)8-s + (−0.854 + 0.519i)9-s + (0.917 − 0.398i)11-s + (−0.136 − 0.990i)12-s + (0.887 + 0.460i)13-s + (0.775 − 0.631i)14-s + (0.962 + 0.269i)16-s + (0.398 − 0.917i)17-s + (0.887 − 0.460i)18-s + (−0.576 − 0.816i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.306 - 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.306 - 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(235\)    =    \(5 \cdot 47\)
Sign: $0.306 - 0.951i$
Analytic conductor: \(1.09133\)
Root analytic conductor: \(1.09133\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{235} (77, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 235,\ (0:\ ),\ 0.306 - 0.951i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5301119486 - 0.3863923690i\)
\(L(\frac12)\) \(\approx\) \(0.5301119486 - 0.3863923690i\)
\(L(1)\) \(\approx\) \(0.6088254653 - 0.2154281926i\)
\(L(1)\) \(\approx\) \(0.6088254653 - 0.2154281926i\)

Euler product

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

−26.5808473739107852071930124749, −25.67095358025517306713690824791, −25.13233967743479177054720357705, −23.48076246089207441498577953014, −22.89928203805145148056599329744, −21.61898820662204536832123141677, −20.71184210541997807724696964271, −19.921394392870070531425933530507, −19.09564943591805607770929544746, −17.7846628395704969487693483897, −16.86171191710357580180523846154, −16.450972219017008811841847162970, −15.2952246035446060835254600572, −14.56512464613741936603654557408, −12.90853072938939639254724458797, −11.67185723407622267881041283605, −10.664247237327059644399291707776, −10.0153320421527176216979585269, −9.1086914048436160149899939919, −8.069377167169354886039436401438, −6.61628313842002534093719923263, −5.89135989810667001601921991492, −4.11165714268063297036323500715, −3.16271721597786462198124676532, −1.19839558267910303992946081741, 0.81405736323316732887550045038, 2.16848176892627558703012400516, 3.325349474685690699011457635972, 5.61187984601198697763041070667, 6.57309300947911068439954456191, 7.23496242773897629056740956385, 8.81003279230685610015640449818, 9.07235814764848668927868726090, 10.76388998725134187092266270084, 11.603700518365490731810581060739, 12.41449056053032254202579944250, 13.49795349253169446122485923858, 14.80942460722356173375754927599, 16.130171339187242027458341757405, 16.74459183206585449958715200331, 17.85151174452880146259427954327, 18.67386110138763524129943648561, 19.24188231846656283277647033503, 20.08752123890972259242904148184, 21.370339134333439657656217146817, 22.41063535496277740519441053095, 23.54317938585879298112044818436, 24.48077552193547573337735524188, 25.416409240063782419892976082861, 25.717222691630476801034059625757

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