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Properties

Label 1-235-235.223-r0-0-0
Degree $1$
Conductor $235$
Sign $-0.352 - 0.935i$
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  + (−0.519 − 0.854i)2-s + (−0.816 − 0.576i)3-s + (−0.460 + 0.887i)4-s + (−0.0682 + 0.997i)6-s + (−0.269 − 0.962i)7-s + (0.997 − 0.0682i)8-s + (0.334 + 0.942i)9-s + (0.990 + 0.136i)11-s + (0.887 − 0.460i)12-s + (0.631 + 0.775i)13-s + (−0.682 + 0.730i)14-s + (−0.576 − 0.816i)16-s + (0.136 + 0.990i)17-s + (0.631 − 0.775i)18-s + (0.203 − 0.979i)19-s + ⋯
L(s)  = 1  + (−0.519 − 0.854i)2-s + (−0.816 − 0.576i)3-s + (−0.460 + 0.887i)4-s + (−0.0682 + 0.997i)6-s + (−0.269 − 0.962i)7-s + (0.997 − 0.0682i)8-s + (0.334 + 0.942i)9-s + (0.990 + 0.136i)11-s + (0.887 − 0.460i)12-s + (0.631 + 0.775i)13-s + (−0.682 + 0.730i)14-s + (−0.576 − 0.816i)16-s + (0.136 + 0.990i)17-s + (0.631 − 0.775i)18-s + (0.203 − 0.979i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.352 - 0.935i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.352 - 0.935i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(235\)    =    \(5 \cdot 47\)
Sign: $-0.352 - 0.935i$
Analytic conductor: \(1.09133\)
Root analytic conductor: \(1.09133\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{235} (223, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 235,\ (0:\ ),\ -0.352 - 0.935i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3885150821 - 0.5613961554i\)
\(L(\frac12)\) \(\approx\) \(0.3885150821 - 0.5613961554i\)
\(L(1)\) \(\approx\) \(0.5501745653 - 0.3834625461i\)
\(L(1)\) \(\approx\) \(0.5501745653 - 0.3834625461i\)

Euler product

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

−26.74753547979939106839141566508, −25.41616453489321074918885126477, −24.97386037895740198052150263390, −23.820876240053979803075742718139, −22.68956558244066338852788856339, −22.41628196859108083606797636833, −21.118964649524658760422050330223, −19.91635337511638599359805404044, −18.60405183083137252084227457081, −18.108962650739243656718127658053, −16.95241974078554168419168583910, −16.292547581668334049991118536147, −15.380972747320201626087335695298, −14.70355376112574658586529952123, −13.29349717729319783116610829511, −11.943860815271280410945681692777, −11.06120237208476715335321801380, −9.735334455411150013579927696382, −9.23046163208088186644588335816, −7.952463054827202636609612676566, −6.52901261226847390571543182732, −5.82701039867459883362593445818, −4.93164236392954505185183917701, −3.45853800390241943839006615598, −1.19561961075539917811674512388, 0.83081631544153292867790366458, 1.91595410028161732598545593208, 3.68392478156804583756568028834, 4.6335179416143137757577730974, 6.43331630050231562581835205616, 7.18285051531276977734632087301, 8.46198327479212476093713669226, 9.64495173554875747528673176524, 10.77634193508314284709387940960, 11.35724851607822983182037389603, 12.437433279891063782705294313385, 13.2625658973394160117531433993, 14.167167687049608875555002109896, 16.07884694304235820887538911169, 17.05798953289486144612762268982, 17.3840572647967292831882742370, 18.66168816211836387138694503821, 19.35123389152486271154720677164, 20.183539646697854234011154572543, 21.37711332798421091280628297267, 22.27229602927760022236131702621, 23.07644140034321778123334783227, 23.94450735146082083386699244340, 25.16042622445371388546631958768, 26.2499317049876745801475044147

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