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Properties

Label 1-235-235.23-r0-0-0
Degree $1$
Conductor $235$
Sign $-0.581 + 0.813i$
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  + (−0.269 − 0.962i)2-s + (−0.887 + 0.460i)3-s + (−0.854 + 0.519i)4-s + (0.682 + 0.730i)6-s + (0.136 + 0.990i)7-s + (0.730 + 0.682i)8-s + (0.576 − 0.816i)9-s + (0.0682 − 0.997i)11-s + (0.519 − 0.854i)12-s + (−0.942 + 0.334i)13-s + (0.917 − 0.398i)14-s + (0.460 − 0.887i)16-s + (−0.997 + 0.0682i)17-s + (−0.942 − 0.334i)18-s + (−0.775 + 0.631i)19-s + ⋯
L(s)  = 1  + (−0.269 − 0.962i)2-s + (−0.887 + 0.460i)3-s + (−0.854 + 0.519i)4-s + (0.682 + 0.730i)6-s + (0.136 + 0.990i)7-s + (0.730 + 0.682i)8-s + (0.576 − 0.816i)9-s + (0.0682 − 0.997i)11-s + (0.519 − 0.854i)12-s + (−0.942 + 0.334i)13-s + (0.917 − 0.398i)14-s + (0.460 − 0.887i)16-s + (−0.997 + 0.0682i)17-s + (−0.942 − 0.334i)18-s + (−0.775 + 0.631i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.581 + 0.813i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.581 + 0.813i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(235\)    =    \(5 \cdot 47\)
Sign: $-0.581 + 0.813i$
Analytic conductor: \(1.09133\)
Root analytic conductor: \(1.09133\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{235} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 235,\ (0:\ ),\ -0.581 + 0.813i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.07281305956 + 0.1414449046i\)
\(L(\frac12)\) \(\approx\) \(0.07281305956 + 0.1414449046i\)
\(L(1)\) \(\approx\) \(0.4802636605 - 0.05991334291i\)
\(L(1)\) \(\approx\) \(0.4802636605 - 0.05991334291i\)

Euler product

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

−25.91783191376886934970818018605, −24.83029458874589082195500475367, −24.12718281550709962000098299341, −23.26417080478943461317521041902, −22.68366910089783905995966661840, −21.705931418442333289796017883206, −20.02756963213911231528375726517, −19.27568144266129565977929925279, −17.971664036436894158057738275174, −17.36998974795186704664256604394, −16.876864561258616919778361525295, −15.62398529751504997983422190320, −14.76500564752070603785849406235, −13.43010914369355147540902145324, −12.86334133188627433583073225950, −11.38669745318697746658852382575, −10.344584083824434436417814309922, −9.43730978854867819051789413172, −7.78741907231093355239456661064, −7.19785281893682429065552346676, −6.32756896087324122364933998592, −4.98432536564823519501477210646, −4.30819710715381119617243307125, −1.80510940618041104099184483737, −0.13797115506502532678362282640, 1.8008893195970295471094737294, 3.18087032431461317100055495866, 4.50871464530142959808504995206, 5.40340734803445352644615004487, 6.70190570329046824088017322420, 8.49429764263847924662316197465, 9.18061702996050673355311373079, 10.41605957589378872396323241636, 11.12564262870986150183930510257, 12.1087103394857153540151373455, 12.694415393093480488192677632692, 14.17491679763461906258549875060, 15.31338033042275593593344279649, 16.56337590407530842041377144057, 17.2338417843745921373857892348, 18.39016081024483251006559467418, 18.91366374942471824826783333553, 20.1667469631858830102333957038, 21.2928279029676495118677846408, 21.89231406235582011494677426106, 22.40601712833558735821507148019, 23.6713326665526246222593105952, 24.60209701742277665351115145858, 26.01310553333252173502820457346, 27.162608316502114056553588354194

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