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Properties

Label 1-431-431.135-r0-0-0
Degree $1$
Conductor $431$
Sign $0.142 - 0.989i$
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  + (−0.0365 − 0.999i)2-s + (−0.457 − 0.889i)3-s + (−0.997 + 0.0729i)4-s + (0.800 + 0.599i)5-s + (−0.872 + 0.489i)6-s + (−0.00730 − 0.999i)7-s + (0.109 + 0.994i)8-s + (−0.581 + 0.813i)9-s + (0.569 − 0.821i)10-s + (0.996 + 0.0875i)11-s + (0.520 + 0.853i)12-s + (0.939 + 0.343i)13-s + (−0.999 + 0.0438i)14-s + (0.167 − 0.985i)15-s + (0.989 − 0.145i)16-s + (0.224 + 0.974i)17-s + ⋯
L(s)  = 1  + (−0.0365 − 0.999i)2-s + (−0.457 − 0.889i)3-s + (−0.997 + 0.0729i)4-s + (0.800 + 0.599i)5-s + (−0.872 + 0.489i)6-s + (−0.00730 − 0.999i)7-s + (0.109 + 0.994i)8-s + (−0.581 + 0.813i)9-s + (0.569 − 0.821i)10-s + (0.996 + 0.0875i)11-s + (0.520 + 0.853i)12-s + (0.939 + 0.343i)13-s + (−0.999 + 0.0438i)14-s + (0.167 − 0.985i)15-s + (0.989 − 0.145i)16-s + (0.224 + 0.974i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 431 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.142 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 431 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.142 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(431\)
Sign: $0.142 - 0.989i$
Analytic conductor: \(2.00155\)
Root analytic conductor: \(2.00155\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{431} (135, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 431,\ (0:\ ),\ 0.142 - 0.989i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9421103169 - 0.8160071850i\)
\(L(\frac12)\) \(\approx\) \(0.9421103169 - 0.8160071850i\)
\(L(1)\) \(\approx\) \(0.8493622653 - 0.5777441876i\)
\(L(1)\) \(\approx\) \(0.8493622653 - 0.5777441876i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad431 \( 1 \)
good2 \( 1 + (-0.0365 - 0.999i)T \)
3 \( 1 + (-0.457 - 0.889i)T \)
5 \( 1 + (0.800 + 0.599i)T \)
7 \( 1 + (-0.00730 - 0.999i)T \)
11 \( 1 + (0.996 + 0.0875i)T \)
13 \( 1 + (0.939 + 0.343i)T \)
17 \( 1 + (0.224 + 0.974i)T \)
19 \( 1 + (-0.295 + 0.955i)T \)
23 \( 1 + (0.724 + 0.688i)T \)
29 \( 1 + (-0.238 - 0.971i)T \)
31 \( 1 + (0.417 + 0.908i)T \)
37 \( 1 + (0.782 - 0.622i)T \)
41 \( 1 + (-0.885 - 0.463i)T \)
43 \( 1 + (0.569 + 0.821i)T \)
47 \( 1 + (0.744 - 0.667i)T \)
53 \( 1 + (-0.994 - 0.102i)T \)
59 \( 1 + (-0.557 + 0.829i)T \)
61 \( 1 + (-0.0657 - 0.997i)T \)
67 \( 1 + (-0.267 + 0.963i)T \)
71 \( 1 + (-0.404 - 0.914i)T \)
73 \( 1 + (-0.483 + 0.875i)T \)
79 \( 1 + (-0.0948 - 0.995i)T \)
83 \( 1 + (0.704 - 0.709i)T \)
89 \( 1 + (-0.773 + 0.634i)T \)
97 \( 1 + (-0.508 - 0.861i)T \)
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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

−24.449720628084384181316034010251, −23.54519706558257479979532261378, −22.34284959531463867654807141661, −22.12725306561828198446591318969, −21.12121946116443887612470471237, −20.27999412908011367889471799947, −18.7643911379147864809433011629, −17.98470353896404776039244872942, −17.15342903306218788561128915109, −16.51090931444743127162312744778, −15.69460511193783889887077133283, −14.946499242645122747028306175808, −14.0283533385511718228787739799, −13.020524943868581569794804411846, −12.01741297016731086803304619749, −10.865088970174829783836902801759, −9.544602589751848471140588477827, −9.12663762113222241317298214248, −8.43217178149573601844042525218, −6.65319477631168125613788112459, −5.982032493997577490732260577899, −5.161662840148392112564805733298, −4.40386629912569128892949069016, −2.99565295154740851064288262103, −0.987918067324936655032182721360, 1.25808208400660361451533991072, 1.76283826821845222631117883779, 3.25480212913473997982900415521, 4.22954962620311210165409209784, 5.74626271848562534293861978811, 6.48536053897282085542758149430, 7.61752213967665049396377893932, 8.75732574086422725606644796779, 9.92109961844170147355617256090, 10.753983878374574761290237316006, 11.37339218308855921130276555754, 12.44944544084210964730631850239, 13.35083283025032540419514020862, 13.91550997158569020225042516617, 14.662684933437493307413273839273, 16.70181109645592573817550681757, 17.26861294064535836977019276829, 17.91056554671276461636215181044, 18.97589167943823923846771802513, 19.35807875981260044890407327246, 20.49848995093182132803333859811, 21.36954355457535825127988313411, 22.21605846786920131406920551927, 23.204181425735088564009418642029, 23.38430677125841334042804901351

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