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Properties

Label 1-431-431.216-r0-0-0
Degree $1$
Conductor $431$
Sign $0.993 - 0.115i$
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  + (−0.934 + 0.357i)2-s + (−0.0365 − 0.999i)3-s + (0.744 − 0.667i)4-s + (0.989 + 0.145i)5-s + (0.391 + 0.920i)6-s + (−0.997 + 0.0729i)7-s + (−0.457 + 0.889i)8-s + (−0.997 + 0.0729i)9-s + (−0.976 + 0.217i)10-s + (0.639 + 0.768i)11-s + (−0.694 − 0.719i)12-s + (−0.934 − 0.357i)13-s + (0.905 − 0.424i)14-s + (0.109 − 0.994i)15-s + (0.109 − 0.994i)16-s + (0.639 + 0.768i)17-s + ⋯
L(s)  = 1  + (−0.934 + 0.357i)2-s + (−0.0365 − 0.999i)3-s + (0.744 − 0.667i)4-s + (0.989 + 0.145i)5-s + (0.391 + 0.920i)6-s + (−0.997 + 0.0729i)7-s + (−0.457 + 0.889i)8-s + (−0.997 + 0.0729i)9-s + (−0.976 + 0.217i)10-s + (0.639 + 0.768i)11-s + (−0.694 − 0.719i)12-s + (−0.934 − 0.357i)13-s + (0.905 − 0.424i)14-s + (0.109 − 0.994i)15-s + (0.109 − 0.994i)16-s + (0.639 + 0.768i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 431 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.993 - 0.115i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 431 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.993 - 0.115i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(431\)
Sign: $0.993 - 0.115i$
Analytic conductor: \(2.00155\)
Root analytic conductor: \(2.00155\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{431} (216, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 431,\ (0:\ ),\ 0.993 - 0.115i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9056480330 - 0.05246227438i\)
\(L(\frac12)\) \(\approx\) \(0.9056480330 - 0.05246227438i\)
\(L(1)\) \(\approx\) \(0.7799717352 - 0.05747591637i\)
\(L(1)\) \(\approx\) \(0.7799717352 - 0.05747591637i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad431 \( 1 \)
good2 \( 1 + (-0.934 + 0.357i)T \)
3 \( 1 + (-0.0365 - 0.999i)T \)
5 \( 1 + (0.989 + 0.145i)T \)
7 \( 1 + (-0.997 + 0.0729i)T \)
11 \( 1 + (0.639 + 0.768i)T \)
13 \( 1 + (-0.934 - 0.357i)T \)
17 \( 1 + (0.639 + 0.768i)T \)
19 \( 1 + (0.989 - 0.145i)T \)
23 \( 1 + (0.252 + 0.967i)T \)
29 \( 1 + (0.744 + 0.667i)T \)
31 \( 1 + (0.391 - 0.920i)T \)
37 \( 1 + (0.905 - 0.424i)T \)
41 \( 1 + (0.109 - 0.994i)T \)
43 \( 1 + (-0.976 - 0.217i)T \)
47 \( 1 + (0.520 - 0.853i)T \)
53 \( 1 + (0.520 + 0.853i)T \)
59 \( 1 + (-0.934 + 0.357i)T \)
61 \( 1 + (-0.791 + 0.611i)T \)
67 \( 1 + (0.905 - 0.424i)T \)
71 \( 1 + (0.520 - 0.853i)T \)
73 \( 1 + (-0.322 + 0.946i)T \)
79 \( 1 + (-0.581 + 0.813i)T \)
83 \( 1 + (-0.0365 - 0.999i)T \)
89 \( 1 + (0.833 - 0.551i)T \)
97 \( 1 + (-0.581 - 0.813i)T \)
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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

−24.76194687869166244017629784514, −22.95896781513199579505715381981, −21.98214240939708355002415117326, −21.63840106584881480152826412022, −20.63057553598499888428301666522, −19.91194274945454148339056336816, −19.047371983526593158893834082491, −18.04628350017512001922834045638, −16.93336258473982976937248096031, −16.59244255375952187910070896679, −15.874472880792571882848338721303, −14.535500078932698017110589507202, −13.68547567393901690899380653881, −12.37355075731301960147625704543, −11.55943161780515677379630253700, −10.408078407304458145955145020, −9.6573135426088612651256246273, −9.37745094918057045314803283842, −8.279219477594599973621196120762, −6.786804819925680620919672452719, −6.00760307416953940217064698420, −4.7178595338810733528581174009, −3.240341694359117006074058937949, −2.66547321949395118813244566081, −0.90047387147572789251295201824, 1.053919396215581945820450780610, 2.110033144676735429702774061834, 3.062571539864477867008281834058, 5.37062778885782334265345396363, 6.08914640111060417224048305241, 6.98498854330086214661724183240, 7.58485910852124528038374722010, 8.9698640296042953630716407369, 9.66904172121880068364192797583, 10.40482049245079784961548791815, 11.822180696969996229383374393677, 12.54868095180783846312537753967, 13.61829936183834000883039912791, 14.51128690127207036595082929400, 15.403236648145178309301745250257, 16.87295313764850223106061319583, 17.13375601083279522412017580573, 18.06593718919804885164479402317, 18.7557866302719806734286574440, 19.7590982924830310085124940079, 20.08923498263984763607710006511, 21.59208671117356202958933287581, 22.58052144723526856435176747211, 23.40080138806911007816156271275, 24.53313046679899671442078515162

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