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Properties

Label 1-431-431.54-r0-0-0
Degree $1$
Conductor $431$
Sign $-0.0947 - 0.995i$
Analytic cond. $2.00155$
Root an. cond. $2.00155$
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  + (−0.457 + 0.889i)2-s + (0.109 + 0.994i)3-s + (−0.581 − 0.813i)4-s + (0.905 + 0.424i)5-s + (−0.934 − 0.357i)6-s + (−0.976 + 0.217i)7-s + (0.989 − 0.145i)8-s + (−0.976 + 0.217i)9-s + (−0.791 + 0.611i)10-s + (−0.872 + 0.489i)11-s + (0.744 − 0.667i)12-s + (−0.457 − 0.889i)13-s + (0.252 − 0.967i)14-s + (−0.322 + 0.946i)15-s + (−0.322 + 0.946i)16-s + (−0.872 + 0.489i)17-s + ⋯
L(s)  = 1  + (−0.457 + 0.889i)2-s + (0.109 + 0.994i)3-s + (−0.581 − 0.813i)4-s + (0.905 + 0.424i)5-s + (−0.934 − 0.357i)6-s + (−0.976 + 0.217i)7-s + (0.989 − 0.145i)8-s + (−0.976 + 0.217i)9-s + (−0.791 + 0.611i)10-s + (−0.872 + 0.489i)11-s + (0.744 − 0.667i)12-s + (−0.457 − 0.889i)13-s + (0.252 − 0.967i)14-s + (−0.322 + 0.946i)15-s + (−0.322 + 0.946i)16-s + (−0.872 + 0.489i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 431 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0947 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 431 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0947 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(431\)
Sign: $-0.0947 - 0.995i$
Analytic conductor: \(2.00155\)
Root analytic conductor: \(2.00155\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{431} (54, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 431,\ (0:\ ),\ -0.0947 - 0.995i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1793788323 + 0.1972638077i\)
\(L(\frac12)\) \(\approx\) \(-0.1793788323 + 0.1972638077i\)
\(L(1)\) \(\approx\) \(0.4115992184 + 0.4563961859i\)
\(L(1)\) \(\approx\) \(0.4115992184 + 0.4563961859i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad431 \( 1 \)
good2 \( 1 + (-0.457 + 0.889i)T \)
3 \( 1 + (0.109 + 0.994i)T \)
5 \( 1 + (0.905 + 0.424i)T \)
7 \( 1 + (-0.976 + 0.217i)T \)
11 \( 1 + (-0.872 + 0.489i)T \)
13 \( 1 + (-0.457 - 0.889i)T \)
17 \( 1 + (-0.872 + 0.489i)T \)
19 \( 1 + (0.905 - 0.424i)T \)
23 \( 1 + (-0.694 - 0.719i)T \)
29 \( 1 + (-0.581 + 0.813i)T \)
31 \( 1 + (-0.934 + 0.357i)T \)
37 \( 1 + (0.252 - 0.967i)T \)
41 \( 1 + (-0.322 + 0.946i)T \)
43 \( 1 + (-0.791 - 0.611i)T \)
47 \( 1 + (-0.997 - 0.0729i)T \)
53 \( 1 + (-0.997 + 0.0729i)T \)
59 \( 1 + (-0.457 + 0.889i)T \)
61 \( 1 + (0.391 + 0.920i)T \)
67 \( 1 + (0.252 - 0.967i)T \)
71 \( 1 + (-0.997 - 0.0729i)T \)
73 \( 1 + (0.833 - 0.551i)T \)
79 \( 1 + (0.957 + 0.288i)T \)
83 \( 1 + (0.109 + 0.994i)T \)
89 \( 1 + (-0.181 - 0.983i)T \)
97 \( 1 + (0.957 - 0.288i)T \)
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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

−23.53444646775180736767115022310, −22.38626390932363437473460235459, −21.80402180637420900807269365715, −20.598831500944146297727417818232, −20.09914523623745011754450107819, −19.06519267319794131441863098338, −18.49411590313801948073787924202, −17.6661296514704008495547030901, −16.82331353641271311925206547176, −16.04910682187913737568436685563, −14.12731671221932906337295779507, −13.43647519126851581887790859943, −13.03013345336854360856288779986, −12.02040221696500671394872657442, −11.16043413993694037387975854881, −9.80356924720480852600788659067, −9.37880024090837827069663806061, −8.25840896011609724480182987003, −7.275132798023121425648742748868, −6.205768096926154256082537526446, −5.05905406713468241171897578068, −3.47831132029944292310402369561, −2.45641888828868746551379132474, −1.63135356624060441240899162096, −0.16193332208963066228393409714, 2.23269978266147689829065093813, 3.35381915400044242501373950901, 4.88200045388046818529288642323, 5.574395512578203953232617544642, 6.47749885086732677453566263501, 7.5742809382949856416051263343, 8.814616337073672399820924147691, 9.58094051450083102918117957977, 10.224193278566941742520484996010, 10.85461507361780039561903150358, 12.770770725105447639697603328442, 13.53831624944713148273058310909, 14.638197349891699918540434635680, 15.24749299292918891468986395491, 16.05844321403321213111971050406, 16.78251499891740639574276705836, 17.881776248477701405772931824166, 18.27502063590665341024935807463, 19.72641250401218398356916558642, 20.25056293963265639855702802959, 21.65064430657511306127748226267, 22.33827004897913114403654491919, 22.783131603938656772224553459650, 24.04419325869910530439946868017, 25.18140519864544129844776808096

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