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Properties

Label 1-235-235.3-r1-0-0
Degree $1$
Conductor $235$
Sign $-0.0973 - 0.995i$
Analytic cond. $25.2542$
Root an. cond. $25.2542$
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  + (−0.887 − 0.460i)2-s + (0.942 − 0.334i)3-s + (0.576 + 0.816i)4-s + (−0.990 − 0.136i)6-s + (−0.519 − 0.854i)7-s + (−0.136 − 0.990i)8-s + (0.775 − 0.631i)9-s + (0.962 + 0.269i)11-s + (0.816 + 0.576i)12-s + (0.979 + 0.203i)13-s + (0.0682 + 0.997i)14-s + (−0.334 + 0.942i)16-s + (−0.269 − 0.962i)17-s + (−0.979 + 0.203i)18-s + (0.917 + 0.398i)19-s + ⋯
L(s)  = 1  + (−0.887 − 0.460i)2-s + (0.942 − 0.334i)3-s + (0.576 + 0.816i)4-s + (−0.990 − 0.136i)6-s + (−0.519 − 0.854i)7-s + (−0.136 − 0.990i)8-s + (0.775 − 0.631i)9-s + (0.962 + 0.269i)11-s + (0.816 + 0.576i)12-s + (0.979 + 0.203i)13-s + (0.0682 + 0.997i)14-s + (−0.334 + 0.942i)16-s + (−0.269 − 0.962i)17-s + (−0.979 + 0.203i)18-s + (0.917 + 0.398i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 235 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.0973 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 235 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.0973 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(235\)    =    \(5 \cdot 47\)
Sign: $-0.0973 - 0.995i$
Analytic conductor: \(25.2542\)
Root analytic conductor: \(25.2542\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{235} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 235,\ (1:\ ),\ -0.0973 - 0.995i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.217610589 - 1.342568413i\)
\(L(\frac12)\) \(\approx\) \(1.217610589 - 1.342568413i\)
\(L(1)\) \(\approx\) \(0.9791189683 - 0.4734512684i\)
\(L(1)\) \(\approx\) \(0.9791189683 - 0.4734512684i\)

Euler product

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

−26.04236416871292112841245633111, −25.64226252307845845993295695284, −24.7029093335604512503921452291, −24.01215388888055808283458921370, −22.446481636610923146890643297461, −21.58936772908312466725428863809, −20.300298422015512965112388915882, −19.73868814508673306432146728975, −18.77071485999442914581634333698, −18.09058834334041023347639101721, −16.64881281739098632623982154075, −15.90903600844632021851643857146, −15.09869498539210485844835629050, −14.26322927908242298388580520109, −13.08647300869670127222126602825, −11.64442115625634602552952537313, −10.4897663452177404217507546696, −9.389733798737582983162004936132, −8.82665742284638770021755159001, −7.94410900069188735517820208281, −6.61018502454563360881275274319, −5.66236522174758636565413048835, −3.917458974422689487254076917528, −2.6282780781534880610847149317, −1.33601681952544392358152387575, 0.78179793536045667315929992544, 1.85767291867474794634679341980, 3.32523866404518747618050255821, 4.00104758607885594455769589579, 6.47233985748210044970014141552, 7.26000436334692436360518239440, 8.23143269075208300595684876403, 9.37466077106899065844631522660, 9.876466326418403404641905098098, 11.2721496724641427823928231460, 12.262286749678678011860733500394, 13.432574695623414703889329018, 14.11116563075880819832618357005, 15.65049386146351229914290831433, 16.38501292960349086823200268647, 17.59581193416485666490445131852, 18.45366310102013997341565084080, 19.351514458464180920053707193951, 20.21653158637732639067575190054, 20.557823592568864685065902706610, 21.85405029321773740993358632644, 22.99720548706234021139528156237, 24.28645471868924867911279090997, 25.24249228340384740753860458516, 25.81577564303204789033749858972

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