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Properties

Label 1-431-431.64-r0-0-0
Degree $1$
Conductor $431$
Sign $0.722 + 0.690i$
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  + (−0.581 + 0.813i)2-s + (−0.976 − 0.217i)3-s + (−0.322 − 0.946i)4-s + (0.639 − 0.768i)5-s + (0.744 − 0.667i)6-s + (0.905 + 0.424i)7-s + (0.957 + 0.288i)8-s + (0.905 + 0.424i)9-s + (0.252 + 0.967i)10-s + (0.520 + 0.853i)11-s + (0.109 + 0.994i)12-s + (−0.581 − 0.813i)13-s + (−0.872 + 0.489i)14-s + (−0.791 + 0.611i)15-s + (−0.791 + 0.611i)16-s + (0.520 + 0.853i)17-s + ⋯
L(s)  = 1  + (−0.581 + 0.813i)2-s + (−0.976 − 0.217i)3-s + (−0.322 − 0.946i)4-s + (0.639 − 0.768i)5-s + (0.744 − 0.667i)6-s + (0.905 + 0.424i)7-s + (0.957 + 0.288i)8-s + (0.905 + 0.424i)9-s + (0.252 + 0.967i)10-s + (0.520 + 0.853i)11-s + (0.109 + 0.994i)12-s + (−0.581 − 0.813i)13-s + (−0.872 + 0.489i)14-s + (−0.791 + 0.611i)15-s + (−0.791 + 0.611i)16-s + (0.520 + 0.853i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 431 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.722 + 0.690i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 431 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.722 + 0.690i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(431\)
Sign: $0.722 + 0.690i$
Analytic conductor: \(2.00155\)
Root analytic conductor: \(2.00155\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{431} (64, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 431,\ (0:\ ),\ 0.722 + 0.690i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8342019192 + 0.3344940551i\)
\(L(\frac12)\) \(\approx\) \(0.8342019192 + 0.3344940551i\)
\(L(1)\) \(\approx\) \(0.7455402270 + 0.1896958916i\)
\(L(1)\) \(\approx\) \(0.7455402270 + 0.1896958916i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad431 \( 1 \)
good2 \( 1 + (-0.581 + 0.813i)T \)
3 \( 1 + (-0.976 - 0.217i)T \)
5 \( 1 + (0.639 - 0.768i)T \)
7 \( 1 + (0.905 + 0.424i)T \)
11 \( 1 + (0.520 + 0.853i)T \)
13 \( 1 + (-0.581 - 0.813i)T \)
17 \( 1 + (0.520 + 0.853i)T \)
19 \( 1 + (0.639 + 0.768i)T \)
23 \( 1 + (-0.0365 - 0.999i)T \)
29 \( 1 + (-0.322 + 0.946i)T \)
31 \( 1 + (0.744 + 0.667i)T \)
37 \( 1 + (-0.872 + 0.489i)T \)
41 \( 1 + (-0.791 + 0.611i)T \)
43 \( 1 + (0.252 - 0.967i)T \)
47 \( 1 + (0.989 - 0.145i)T \)
53 \( 1 + (0.989 + 0.145i)T \)
59 \( 1 + (-0.581 + 0.813i)T \)
61 \( 1 + (-0.694 - 0.719i)T \)
67 \( 1 + (-0.872 + 0.489i)T \)
71 \( 1 + (0.989 - 0.145i)T \)
73 \( 1 + (0.391 + 0.920i)T \)
79 \( 1 + (0.833 - 0.551i)T \)
83 \( 1 + (-0.976 - 0.217i)T \)
89 \( 1 + (-0.934 - 0.357i)T \)
97 \( 1 + (0.833 + 0.551i)T \)
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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

−24.00457225179731604198670133540, −22.78742735513211156110540522080, −22.182789230643216700728648499255, −21.3256296001678080520728052446, −20.97130015738218026905220020354, −19.55262998077814946858705187272, −18.695020977912705270735586744472, −17.97117601935888352177289669317, −17.19565376536314116674343534194, −16.74090925742299546660616431447, −15.45672273623305669338739078331, −14.00549771046552137969336784801, −13.58614780838373518636528993603, −11.94550993055185091577805634825, −11.48985512322884743520068712187, −10.82566671688743000128595554713, −9.827772343408479049731298307370, −9.199958298197112827815756558063, −7.6092238072813392481310525420, −6.915410125412246165609894883224, −5.59192555776327632859526156542, −4.53032165167835047726944814055, −3.41365111869747562566244318003, −2.0562039149560182676535150108, −0.91822590370152827950640405704, 1.14701840235157509148770440406, 1.87514490662062000019823413149, 4.47757970627245327470307650459, 5.22884197840210902715036228968, 5.831397337030664773181170500345, 6.94964125364658591803295957900, 7.946997611882240407153883682233, 8.820610892832332830168183237, 10.04409361783480760452865214899, 10.505197872651331545212022692197, 12.05239777375346502693942604349, 12.53519218656336209203334190053, 13.8705911425281367469177259436, 14.79468253517010337932929291236, 15.66187934906358503524273710568, 16.92044646348040197060038628179, 17.074627492247901735246564453382, 17.999316737268237151948819724365, 18.53966265834759992183624210836, 19.87012944704958610410801226393, 20.753523583801621698600797363447, 21.90297584761079265531719996022, 22.66658129255990282821777938639, 23.624708859241570699598589955172, 24.467227310176301464265075072335

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