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Properties

Label 1-431-431.12-r0-0-0
Degree $1$
Conductor $431$
Sign $-0.802 - 0.596i$
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  + (−0.694 + 0.719i)2-s + (0.520 − 0.853i)3-s + (−0.0365 − 0.999i)4-s + (−0.581 − 0.813i)5-s + (0.252 + 0.967i)6-s + (−0.457 − 0.889i)7-s + (0.744 + 0.667i)8-s + (−0.457 − 0.889i)9-s + (0.989 + 0.145i)10-s + (0.833 − 0.551i)11-s + (−0.872 − 0.489i)12-s + (−0.694 − 0.719i)13-s + (0.957 + 0.288i)14-s + (−0.997 + 0.0729i)15-s + (−0.997 + 0.0729i)16-s + (0.833 − 0.551i)17-s + ⋯
L(s)  = 1  + (−0.694 + 0.719i)2-s + (0.520 − 0.853i)3-s + (−0.0365 − 0.999i)4-s + (−0.581 − 0.813i)5-s + (0.252 + 0.967i)6-s + (−0.457 − 0.889i)7-s + (0.744 + 0.667i)8-s + (−0.457 − 0.889i)9-s + (0.989 + 0.145i)10-s + (0.833 − 0.551i)11-s + (−0.872 − 0.489i)12-s + (−0.694 − 0.719i)13-s + (0.957 + 0.288i)14-s + (−0.997 + 0.0729i)15-s + (−0.997 + 0.0729i)16-s + (0.833 − 0.551i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 431 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.802 - 0.596i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 431 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.802 - 0.596i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(431\)
Sign: $-0.802 - 0.596i$
Analytic conductor: \(2.00155\)
Root analytic conductor: \(2.00155\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{431} (12, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 431,\ (0:\ ),\ -0.802 - 0.596i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2363243408 - 0.7141661135i\)
\(L(\frac12)\) \(\approx\) \(0.2363243408 - 0.7141661135i\)
\(L(1)\) \(\approx\) \(0.6617202244 - 0.3307327612i\)
\(L(1)\) \(\approx\) \(0.6617202244 - 0.3307327612i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad431 \( 1 \)
good2 \( 1 + (-0.694 + 0.719i)T \)
3 \( 1 + (0.520 - 0.853i)T \)
5 \( 1 + (-0.581 - 0.813i)T \)
7 \( 1 + (-0.457 - 0.889i)T \)
11 \( 1 + (0.833 - 0.551i)T \)
13 \( 1 + (-0.694 - 0.719i)T \)
17 \( 1 + (0.833 - 0.551i)T \)
19 \( 1 + (-0.581 + 0.813i)T \)
23 \( 1 + (0.639 - 0.768i)T \)
29 \( 1 + (-0.0365 + 0.999i)T \)
31 \( 1 + (0.252 - 0.967i)T \)
37 \( 1 + (0.957 + 0.288i)T \)
41 \( 1 + (-0.997 + 0.0729i)T \)
43 \( 1 + (0.989 - 0.145i)T \)
47 \( 1 + (-0.934 + 0.357i)T \)
53 \( 1 + (-0.934 - 0.357i)T \)
59 \( 1 + (-0.694 + 0.719i)T \)
61 \( 1 + (0.905 + 0.424i)T \)
67 \( 1 + (0.957 + 0.288i)T \)
71 \( 1 + (-0.934 + 0.357i)T \)
73 \( 1 + (-0.976 + 0.217i)T \)
79 \( 1 + (0.109 - 0.994i)T \)
83 \( 1 + (0.520 - 0.853i)T \)
89 \( 1 + (-0.791 + 0.611i)T \)
97 \( 1 + (0.109 + 0.994i)T \)
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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

−25.00808789912094316644992864959, −23.306258461973123834230972820978, −22.320844423003037273727513521325, −21.7730545315099897563416556914, −21.16282153105131651665811490488, −19.85804880319550663825537719487, −19.336910165007368474714971931498, −18.898169353827902392717677252484, −17.56050360301006367314391818269, −16.7451597571978146834631364221, −15.69461144652803672553526505463, −15.01240530744140592306453567662, −14.13387935667775735313700435236, −12.73963621764870973948085284293, −11.78660077900904499241606012264, −11.136768851792826381216008853241, −9.97874764833906955858609055155, −9.44879578389928213321393834953, −8.5600571108622218383359450853, −7.53337390839877365637039881723, −6.49467610951773214853355806806, −4.7285092843052501558819217275, −3.74111386426287222350874392548, −2.91400375760953580781268439091, −2.00667137672692575766505908455, 0.5431799701729854705894312143, 1.37373987842415358179429146146, 3.12589819235311956847583845480, 4.39175212356709764274013243266, 5.73945499272671857731310303860, 6.76318140596878332141064562906, 7.60303566009920567346948897219, 8.2494434180258114720669383160, 9.16603519764286032894709073480, 10.0554574979337749888094053130, 11.360556176116581183950472014131, 12.464208783431563834625103846173, 13.27660125568943657835295190977, 14.34457541690779131455700005058, 14.917698002563938279196822884, 16.30039424270902814157149364793, 16.79479029358894313828706668671, 17.55267205875731508699519735674, 18.88840990858118264644620622412, 19.2304782685287097844485442589, 20.19369494658663293687797007705, 20.57927278338756670627418605080, 22.541086909729063866506899865828, 23.30822961674962681172805848960, 23.96840340166841197360175531831

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