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Properties

Label 1-731-731.156-r0-0-0
Degree $1$
Conductor $731$
Sign $0.673 + 0.739i$
Analytic cond. $3.39474$
Root an. cond. $3.39474$
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  + (0.330 − 0.943i)2-s + (0.666 + 0.745i)3-s + (−0.781 − 0.623i)4-s + (0.815 + 0.578i)5-s + (0.923 − 0.382i)6-s + (0.382 + 0.923i)7-s + (−0.846 + 0.532i)8-s + (−0.111 + 0.993i)9-s + (0.815 − 0.578i)10-s + (−0.875 − 0.483i)11-s + (−0.0560 − 0.998i)12-s + (−0.974 − 0.222i)13-s + (0.998 − 0.0560i)14-s + (0.111 + 0.993i)15-s + (0.222 + 0.974i)16-s + ⋯
L(s)  = 1  + (0.330 − 0.943i)2-s + (0.666 + 0.745i)3-s + (−0.781 − 0.623i)4-s + (0.815 + 0.578i)5-s + (0.923 − 0.382i)6-s + (0.382 + 0.923i)7-s + (−0.846 + 0.532i)8-s + (−0.111 + 0.993i)9-s + (0.815 − 0.578i)10-s + (−0.875 − 0.483i)11-s + (−0.0560 − 0.998i)12-s + (−0.974 − 0.222i)13-s + (0.998 − 0.0560i)14-s + (0.111 + 0.993i)15-s + (0.222 + 0.974i)16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.673 + 0.739i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.673 + 0.739i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(731\)    =    \(17 \cdot 43\)
Sign: $0.673 + 0.739i$
Analytic conductor: \(3.39474\)
Root analytic conductor: \(3.39474\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{731} (156, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 731,\ (0:\ ),\ 0.673 + 0.739i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.785140443 + 0.7889384068i\)
\(L(\frac12)\) \(\approx\) \(1.785140443 + 0.7889384068i\)
\(L(1)\) \(\approx\) \(1.478645462 + 0.07033444926i\)
\(L(1)\) \(\approx\) \(1.478645462 + 0.07033444926i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 \)
43 \( 1 \)
good2 \( 1 + (0.330 - 0.943i)T \)
3 \( 1 + (0.666 + 0.745i)T \)
5 \( 1 + (0.815 + 0.578i)T \)
7 \( 1 + (0.382 + 0.923i)T \)
11 \( 1 + (-0.875 - 0.483i)T \)
13 \( 1 + (-0.974 - 0.222i)T \)
19 \( 1 + (0.111 + 0.993i)T \)
23 \( 1 + (0.875 + 0.483i)T \)
29 \( 1 + (-0.0560 - 0.998i)T \)
31 \( 1 + (0.998 - 0.0560i)T \)
37 \( 1 + (-0.923 - 0.382i)T \)
41 \( 1 + (0.745 + 0.666i)T \)
47 \( 1 + (0.781 + 0.623i)T \)
53 \( 1 + (-0.846 - 0.532i)T \)
59 \( 1 + (-0.532 + 0.846i)T \)
61 \( 1 + (-0.0560 + 0.998i)T \)
67 \( 1 + (-0.623 + 0.781i)T \)
71 \( 1 + (-0.875 + 0.483i)T \)
73 \( 1 + (0.815 + 0.578i)T \)
79 \( 1 + (-0.923 + 0.382i)T \)
83 \( 1 + (0.943 - 0.330i)T \)
89 \( 1 + (0.433 - 0.900i)T \)
97 \( 1 + (0.960 - 0.276i)T \)
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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

−22.63324845801260424489685511933, −21.5491597573445066369400425344, −20.80068273575757587758275625869, −20.167035620014498357656003389437, −19.055906305249843139988561430098, −18.05796255921232763294645633288, −17.411068103142898532002409940552, −16.96982209296826148701441552753, −15.77111575801481396175127166500, −14.871332776481865437540673626006, −14.0636224884849787959120834031, −13.54988172323745386174295751791, −12.800425569651123987846070300238, −12.179225769276857818226917077699, −10.54051061261804929336814604466, −9.48709848746757364112905909950, −8.79736196215391893969068717237, −7.812895616280241792687006829182, −7.16003349553876080609507829622, −6.44255769308828974233728816698, −5.0659020826071802685454345896, −4.64215061316923856625927796891, −3.166301492451775079631168712614, −2.13172032523109270240437706526, −0.74165757463862397628123244050, 1.7216874774309230336477795085, 2.69325888081845517656576200789, 2.996231849132512201020895809607, 4.40798309445107610848241982533, 5.3579664541433061910761039490, 5.865534673018444253083289591766, 7.659947227977376386968051915963, 8.619500436532461221985477818280, 9.49400336349825924323360875307, 10.1142227135761942617094403275, 10.822437842931854336977845627852, 11.73313802372430883635155937187, 12.8187121259610554165191739872, 13.63088752909762841945811464791, 14.393765646061949624782943058617, 14.982538687358896429923973285177, 15.74500559755801499735785040782, 17.15135868123299342722497507556, 17.95538955817117091879272875946, 19.00302637707592592934748633897, 19.19312067438373597895848838025, 20.59008238927747474086592037234, 21.150042637636539159621898159132, 21.52430400915381987892121725938, 22.36111988921513805149571981730

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