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Properties

Label 1-235-235.138-r0-0-0
Degree $1$
Conductor $235$
Sign $-0.611 - 0.791i$
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  + (−0.887 − 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

\[\begin{aligned}\Lambda(s)=\mathstrut & 235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.611 - 0.791i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.611 - 0.791i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(235\)    =    \(5 \cdot 47\)
Sign: $-0.611 - 0.791i$
Analytic conductor: \(1.09133\)
Root analytic conductor: \(1.09133\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{235} (138, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 235,\ (0:\ ),\ -0.611 - 0.791i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3594680290 - 0.7320217546i\)
\(L(\frac12)\) \(\approx\) \(0.3594680290 - 0.7320217546i\)
\(L(1)\) \(\approx\) \(0.6946153422 - 0.4257193024i\)
\(L(1)\) \(\approx\) \(0.6946153422 - 0.4257193024i\)

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

−26.5336526035096752196932909595, −25.67253405813114116980201687595, −25.11362608777469824534434064254, −24.20997012242352324837782064970, −23.10464173206806287825903307809, −21.63347020624158175808516481396, −20.99115960781102026930878830213, −19.65377085806845799925263383909, −19.25571342383733977727907021751, −18.33806119417248693433774943061, −17.164271846167716345302581288542, −16.112799957634240245276055480543, −15.174905634862480050354482331958, −14.837668267900988487978534369618, −13.36042583733865147358632831990, −12.27592233124704596348197846399, −10.67840077596888845400542968992, −9.89732637822329048692804430579, −8.99771735426408832809754162669, −8.17208127885992137729657183705, −7.19307012680333386423950273586, −5.90678411087073895916433474016, −4.63815920371443026127185065609, −2.82765694141884902721362943871, −2.00946109779053033464679418432, 0.66394862927028391007343991541, 2.39749833005644979938326836821, 3.09072758159059182911620083204, 4.503708884119964300116298623435, 6.69951390653322910780780016775, 7.42592757596907119931384763557, 8.340105419023955546068718140765, 9.45072548269564224397628276373, 10.190635575939519102187527287035, 11.27103532423744330713403822636, 12.81907428999769631282194381567, 13.136173530379696395428518774641, 14.54400460913576155351037946895, 15.689198400384165842728781421609, 16.6326539172975375991662864050, 17.70325708328086617330399140270, 18.67206936115961822666695345817, 19.43520085697699792461001781874, 20.19255087474747270049756452368, 20.89256025603566761887342488327, 21.92289700373644812491836467163, 23.30439481088143977084775603078, 24.41188376087928359952765610249, 25.257661204482370866713186590918, 26.1969929497305713568074488478

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