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Properties

Label 1-431-431.72-r0-0-0
Degree $1$
Conductor $431$
Sign $0.666 + 0.745i$
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  + (0.391 + 0.920i)2-s + (−0.872 + 0.489i)3-s + (−0.694 + 0.719i)4-s + (−0.457 + 0.889i)5-s + (−0.791 − 0.611i)6-s + (0.520 − 0.853i)7-s + (−0.934 − 0.357i)8-s + (0.520 − 0.853i)9-s + (−0.997 − 0.0729i)10-s + (0.957 − 0.288i)11-s + (0.252 − 0.967i)12-s + (0.391 − 0.920i)13-s + (0.989 + 0.145i)14-s + (−0.0365 − 0.999i)15-s + (−0.0365 − 0.999i)16-s + (0.957 − 0.288i)17-s + ⋯
L(s)  = 1  + (0.391 + 0.920i)2-s + (−0.872 + 0.489i)3-s + (−0.694 + 0.719i)4-s + (−0.457 + 0.889i)5-s + (−0.791 − 0.611i)6-s + (0.520 − 0.853i)7-s + (−0.934 − 0.357i)8-s + (0.520 − 0.853i)9-s + (−0.997 − 0.0729i)10-s + (0.957 − 0.288i)11-s + (0.252 − 0.967i)12-s + (0.391 − 0.920i)13-s + (0.989 + 0.145i)14-s + (−0.0365 − 0.999i)15-s + (−0.0365 − 0.999i)16-s + (0.957 − 0.288i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 431 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.666 + 0.745i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 431 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.666 + 0.745i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(431\)
Sign: $0.666 + 0.745i$
Analytic conductor: \(2.00155\)
Root analytic conductor: \(2.00155\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{431} (72, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 431,\ (0:\ ),\ 0.666 + 0.745i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9662338237 + 0.4320936949i\)
\(L(\frac12)\) \(\approx\) \(0.9662338237 + 0.4320936949i\)
\(L(1)\) \(\approx\) \(0.8244965691 + 0.4543034262i\)
\(L(1)\) \(\approx\) \(0.8244965691 + 0.4543034262i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad431 \( 1 \)
good2 \( 1 + (0.391 + 0.920i)T \)
3 \( 1 + (-0.872 + 0.489i)T \)
5 \( 1 + (-0.457 + 0.889i)T \)
7 \( 1 + (0.520 - 0.853i)T \)
11 \( 1 + (0.957 - 0.288i)T \)
13 \( 1 + (0.391 - 0.920i)T \)
17 \( 1 + (0.957 - 0.288i)T \)
19 \( 1 + (-0.457 - 0.889i)T \)
23 \( 1 + (0.905 - 0.424i)T \)
29 \( 1 + (-0.694 - 0.719i)T \)
31 \( 1 + (-0.791 + 0.611i)T \)
37 \( 1 + (0.989 + 0.145i)T \)
41 \( 1 + (-0.0365 - 0.999i)T \)
43 \( 1 + (-0.997 + 0.0729i)T \)
47 \( 1 + (-0.181 - 0.983i)T \)
53 \( 1 + (-0.181 + 0.983i)T \)
59 \( 1 + (0.391 + 0.920i)T \)
61 \( 1 + (-0.976 - 0.217i)T \)
67 \( 1 + (0.989 + 0.145i)T \)
71 \( 1 + (-0.181 - 0.983i)T \)
73 \( 1 + (0.109 + 0.994i)T \)
79 \( 1 + (0.744 - 0.667i)T \)
83 \( 1 + (-0.872 + 0.489i)T \)
89 \( 1 + (-0.322 - 0.946i)T \)
97 \( 1 + (0.744 + 0.667i)T \)
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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

−23.70352619753832629291601872327, −23.285754619107209242271964054788, −22.22467750163696238573004941059, −21.438648878931523104270456381600, −20.77130731612483236403761235386, −19.61657950344085793930031153913, −18.891478182615746925299948399987, −18.252823809468315561877505384125, −17.06243606617450696996885307708, −16.45787065062549188333460819825, −15.06900377669173668442076022281, −14.27046525842821102866197998602, −12.90211867534665493171400855353, −12.50856734295016840110753292417, −11.48983387141587631525925436272, −11.33482806677574456499611015775, −9.74529841765283770734742329960, −8.907303007314542251385014638647, −7.85762976869597416916032191117, −6.33682041611659296328503411558, −5.45380581638382815367033160690, −4.638870879007723975515490844986, −3.668544605867828385499355305909, −1.80403038370584194675872445519, −1.30561560302372750990053885508, 0.71227535084211141423132652443, 3.27390504668661072967491049774, 3.97488366739698638452295529, 4.96766350651138172225207399244, 6.02202259453369246655452808653, 6.90171042690845747997485478697, 7.59263579563306685666111141792, 8.83345687201799350539485681822, 10.09057110096404084601317082828, 11.01888987644141686087983773622, 11.72552752824830606476976431536, 12.873952280974710083134195353000, 13.97942419261092744371485541182, 14.87020171697986311093773253150, 15.36731948218567830481664299982, 16.55938034882662961317357648850, 17.04809351645063827047653481370, 17.93344595385399785787203496001, 18.69674299865406242639200582033, 20.06988539656985376257322364252, 21.19122780951890907248807797453, 21.97140446559804446489454856977, 22.81934413226768479564127472309, 23.21858043716914911455807749582, 23.97487370951278081874138453541

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