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Properties

Label 1-431-431.174-r0-0-0
Degree $1$
Conductor $431$
Sign $0.397 - 0.917i$
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  + (−0.322 − 0.946i)2-s + (0.905 + 0.424i)3-s + (−0.791 + 0.611i)4-s + (0.879 − 0.476i)5-s + (0.109 − 0.994i)6-s + (−0.0657 − 0.997i)7-s + (0.833 + 0.551i)8-s + (0.639 + 0.768i)9-s + (−0.734 − 0.678i)10-s + (0.704 + 0.709i)11-s + (−0.976 + 0.217i)12-s + (−0.999 − 0.0146i)13-s + (−0.923 + 0.384i)14-s + (0.998 − 0.0584i)15-s + (0.252 − 0.967i)16-s + (0.892 − 0.450i)17-s + ⋯
L(s)  = 1  + (−0.322 − 0.946i)2-s + (0.905 + 0.424i)3-s + (−0.791 + 0.611i)4-s + (0.879 − 0.476i)5-s + (0.109 − 0.994i)6-s + (−0.0657 − 0.997i)7-s + (0.833 + 0.551i)8-s + (0.639 + 0.768i)9-s + (−0.734 − 0.678i)10-s + (0.704 + 0.709i)11-s + (−0.976 + 0.217i)12-s + (−0.999 − 0.0146i)13-s + (−0.923 + 0.384i)14-s + (0.998 − 0.0584i)15-s + (0.252 − 0.967i)16-s + (0.892 − 0.450i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 431 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.397 - 0.917i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 431 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.397 - 0.917i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(431\)
Sign: $0.397 - 0.917i$
Analytic conductor: \(2.00155\)
Root analytic conductor: \(2.00155\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{431} (174, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 431,\ (0:\ ),\ 0.397 - 0.917i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.431141343 - 0.9398440959i\)
\(L(\frac12)\) \(\approx\) \(1.431141343 - 0.9398440959i\)
\(L(1)\) \(\approx\) \(1.223228488 - 0.5307356594i\)
\(L(1)\) \(\approx\) \(1.223228488 - 0.5307356594i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad431 \( 1 \)
good2 \( 1 + (-0.322 - 0.946i)T \)
3 \( 1 + (0.905 + 0.424i)T \)
5 \( 1 + (0.879 - 0.476i)T \)
7 \( 1 + (-0.0657 - 0.997i)T \)
11 \( 1 + (0.704 + 0.709i)T \)
13 \( 1 + (-0.999 - 0.0146i)T \)
17 \( 1 + (0.892 - 0.450i)T \)
19 \( 1 + (-0.431 - 0.902i)T \)
23 \( 1 + (0.849 + 0.527i)T \)
29 \( 1 + (-0.825 + 0.563i)T \)
31 \( 1 + (-0.672 - 0.739i)T \)
37 \( 1 + (0.972 + 0.231i)T \)
41 \( 1 + (0.364 + 0.931i)T \)
43 \( 1 + (-0.734 + 0.678i)T \)
47 \( 1 + (0.957 - 0.288i)T \)
53 \( 1 + (-0.605 - 0.795i)T \)
59 \( 1 + (0.817 + 0.575i)T \)
61 \( 1 + (-0.557 - 0.829i)T \)
67 \( 1 + (-0.650 - 0.759i)T \)
71 \( 1 + (0.569 + 0.821i)T \)
73 \( 1 + (0.984 - 0.174i)T \)
79 \( 1 + (-0.754 - 0.656i)T \)
83 \( 1 + (0.683 - 0.729i)T \)
89 \( 1 + (-0.994 - 0.102i)T \)
97 \( 1 + (0.996 - 0.0875i)T \)
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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

−24.65950901082966999658570410891, −23.81336758609093817206909498647, −22.556429438455317424935235144583, −21.80695050884882351688418918322, −21.00420903697493507363078670257, −19.55752384930124468910587544592, −18.82612116919717386363014911391, −18.47720316142653949440839810179, −17.29847174478065017129919083889, −16.61697039621384164518184331918, −15.25809553692209578578361633229, −14.57166253843505019269295309914, −14.22181221997381975244542232504, −13.061348562210503372767830822785, −12.25114450965545048502000535945, −10.56325272501686108552460917403, −9.5199578362299879851987698137, −9.02033920365959942908611692844, −8.05732769127585432253160386832, −7.0498107822664319140028306222, −6.14774642906276671438901362149, −5.41686186755221982609464103700, −3.78707325024972747343481371268, −2.53631220336969226606274176926, −1.44274146863021857596668510709, 1.19544147111955542144390806456, 2.19951735191747176789977736714, 3.25700706311444651954156926943, 4.38891147855446782250513944335, 5.048709424235405707504808061265, 7.05858108614571456342805334500, 7.865037797422395990282435914686, 9.259674754006039258527592318107, 9.51473406967138691319028617971, 10.30275479574539035862305018890, 11.37949273143419636498352482948, 12.76784451065808497893701705744, 13.2255055849008913983233517853, 14.2171959420672849067138072346, 14.85181240049921078208092925118, 16.64277740133425067133117531948, 16.95018311939874096018196999716, 17.961486116889511752501795980416, 19.11024022480714669390687085916, 20.044314386703235079582246382240, 20.26555944356444818823094539950, 21.272952437828485211563546520005, 21.87910571663321582919002447115, 22.7996667359887942138475972882, 24.030684260723769660220799135594

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