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Properties

Label 1-431-431.55-r0-0-0
Degree $1$
Conductor $431$
Sign $-0.981 + 0.193i$
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  + (0.520 − 0.853i)2-s + (0.744 − 0.667i)3-s + (−0.457 − 0.889i)4-s + (−0.976 + 0.217i)5-s + (−0.181 − 0.983i)6-s + (0.109 − 0.994i)7-s + (−0.997 − 0.0729i)8-s + (0.109 − 0.994i)9-s + (−0.322 + 0.946i)10-s + (0.252 − 0.967i)11-s + (−0.934 − 0.357i)12-s + (0.520 + 0.853i)13-s + (−0.791 − 0.611i)14-s + (−0.581 + 0.813i)15-s + (−0.581 + 0.813i)16-s + (0.252 − 0.967i)17-s + ⋯
L(s)  = 1  + (0.520 − 0.853i)2-s + (0.744 − 0.667i)3-s + (−0.457 − 0.889i)4-s + (−0.976 + 0.217i)5-s + (−0.181 − 0.983i)6-s + (0.109 − 0.994i)7-s + (−0.997 − 0.0729i)8-s + (0.109 − 0.994i)9-s + (−0.322 + 0.946i)10-s + (0.252 − 0.967i)11-s + (−0.934 − 0.357i)12-s + (0.520 + 0.853i)13-s + (−0.791 − 0.611i)14-s + (−0.581 + 0.813i)15-s + (−0.581 + 0.813i)16-s + (0.252 − 0.967i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 431 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.981 + 0.193i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 431 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.981 + 0.193i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(431\)
Sign: $-0.981 + 0.193i$
Analytic conductor: \(2.00155\)
Root analytic conductor: \(2.00155\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{431} (55, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 431,\ (0:\ ),\ -0.981 + 0.193i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1500569397 - 1.537615080i\)
\(L(\frac12)\) \(\approx\) \(-0.1500569397 - 1.537615080i\)
\(L(1)\) \(\approx\) \(0.7645158794 - 1.087685410i\)
\(L(1)\) \(\approx\) \(0.7645158794 - 1.087685410i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad431 \( 1 \)
good2 \( 1 + (0.520 - 0.853i)T \)
3 \( 1 + (0.744 - 0.667i)T \)
5 \( 1 + (-0.976 + 0.217i)T \)
7 \( 1 + (0.109 - 0.994i)T \)
11 \( 1 + (0.252 - 0.967i)T \)
13 \( 1 + (0.520 + 0.853i)T \)
17 \( 1 + (0.252 - 0.967i)T \)
19 \( 1 + (-0.976 - 0.217i)T \)
23 \( 1 + (0.391 + 0.920i)T \)
29 \( 1 + (-0.457 + 0.889i)T \)
31 \( 1 + (-0.181 + 0.983i)T \)
37 \( 1 + (-0.791 - 0.611i)T \)
41 \( 1 + (-0.581 + 0.813i)T \)
43 \( 1 + (-0.322 - 0.946i)T \)
47 \( 1 + (-0.0365 - 0.999i)T \)
53 \( 1 + (-0.0365 + 0.999i)T \)
59 \( 1 + (0.520 - 0.853i)T \)
61 \( 1 + (0.833 - 0.551i)T \)
67 \( 1 + (-0.791 - 0.611i)T \)
71 \( 1 + (-0.0365 - 0.999i)T \)
73 \( 1 + (0.957 + 0.288i)T \)
79 \( 1 + (0.989 - 0.145i)T \)
83 \( 1 + (0.744 - 0.667i)T \)
89 \( 1 + (0.639 + 0.768i)T \)
97 \( 1 + (0.989 + 0.145i)T \)
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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

−24.666523997900007284341644818470, −23.84737562680836059520847683104, −22.65135776917197944914210312875, −22.382309074338483157261256785607, −21.00839596983157171804963727948, −20.68281318062372085529260664461, −19.406648672657399633373145074869, −18.637104897035951319313546556267, −17.37387243893137781971452956897, −16.47033874753832787895361256607, −15.5058354275726581196311696380, −15.05178305984805369646657958647, −14.638536585633245092731135397, −13.079954125522665860290724329764, −12.568165573116549407786863911144, −11.47197660844502821294535649711, −10.16313522541148607631597381150, −8.89454854486169162342566145778, −8.3429935604231209742818719539, −7.63896955730267794421766715584, −6.29009785931520445797863844381, −5.13529908696600778972477022992, −4.260154571711477345967814064759, −3.49735023563435887685472308436, −2.29375888193687935773302411803, 0.69089699692584366668736982883, 1.82559585175141035395771416971, 3.42944499677935083532740424559, 3.57812875657023681630874025687, 4.88237682086452969421186995575, 6.51581618609265583815995197526, 7.228845718942163832152979447689, 8.48423329423995690526801475460, 9.19988554167573763364901857129, 10.61835645416253367498093458495, 11.350553871146297193310494817194, 12.12150100021115268217732269886, 13.22125635356911196004002551364, 13.90633783103256747383358685065, 14.49915903220770017461811125281, 15.55323040896201206294484270279, 16.66511325392968064489677563652, 18.07411107002622377628216381660, 18.90947662531703917354237442852, 19.43536046470755685946915490130, 20.13323048364830740627153451348, 20.9241530629267632656534830277, 21.80936462624516290131296653625, 23.14892301449589608306521548632, 23.56207827682876988402050782651

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