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Properties

Label 1-731-731.65-r0-0-0
Degree $1$
Conductor $731$
Sign $-0.998 - 0.0514i$
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  + (−0.993 − 0.111i)2-s + (0.875 − 0.483i)3-s + (0.974 + 0.222i)4-s + (−0.0560 − 0.998i)5-s + (−0.923 + 0.382i)6-s + (−0.382 − 0.923i)7-s + (−0.943 − 0.330i)8-s + (0.532 − 0.846i)9-s + (−0.0560 + 0.998i)10-s + (−0.578 − 0.815i)11-s + (0.960 − 0.276i)12-s + (−0.433 − 0.900i)13-s + (0.276 + 0.960i)14-s + (−0.532 − 0.846i)15-s + (0.900 + 0.433i)16-s + ⋯
L(s)  = 1  + (−0.993 − 0.111i)2-s + (0.875 − 0.483i)3-s + (0.974 + 0.222i)4-s + (−0.0560 − 0.998i)5-s + (−0.923 + 0.382i)6-s + (−0.382 − 0.923i)7-s + (−0.943 − 0.330i)8-s + (0.532 − 0.846i)9-s + (−0.0560 + 0.998i)10-s + (−0.578 − 0.815i)11-s + (0.960 − 0.276i)12-s + (−0.433 − 0.900i)13-s + (0.276 + 0.960i)14-s + (−0.532 − 0.846i)15-s + (0.900 + 0.433i)16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0514i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0514i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(731\)    =    \(17 \cdot 43\)
Sign: $-0.998 - 0.0514i$
Analytic conductor: \(3.39474\)
Root analytic conductor: \(3.39474\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{731} (65, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 731,\ (0:\ ),\ -0.998 - 0.0514i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.02399194084 - 0.9312905357i\)
\(L(\frac12)\) \(\approx\) \(0.02399194084 - 0.9312905357i\)
\(L(1)\) \(\approx\) \(0.6263880865 - 0.5297898860i\)
\(L(1)\) \(\approx\) \(0.6263880865 - 0.5297898860i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 \)
43 \( 1 \)
good2 \( 1 + (-0.993 - 0.111i)T \)
3 \( 1 + (0.875 - 0.483i)T \)
5 \( 1 + (-0.0560 - 0.998i)T \)
7 \( 1 + (-0.382 - 0.923i)T \)
11 \( 1 + (-0.578 - 0.815i)T \)
13 \( 1 + (-0.433 - 0.900i)T \)
19 \( 1 + (-0.532 - 0.846i)T \)
23 \( 1 + (0.578 + 0.815i)T \)
29 \( 1 + (0.960 - 0.276i)T \)
31 \( 1 + (0.276 + 0.960i)T \)
37 \( 1 + (0.923 + 0.382i)T \)
41 \( 1 + (-0.483 + 0.875i)T \)
47 \( 1 + (-0.974 - 0.222i)T \)
53 \( 1 + (-0.943 + 0.330i)T \)
59 \( 1 + (0.330 + 0.943i)T \)
61 \( 1 + (0.960 + 0.276i)T \)
67 \( 1 + (0.222 - 0.974i)T \)
71 \( 1 + (-0.578 + 0.815i)T \)
73 \( 1 + (-0.0560 - 0.998i)T \)
79 \( 1 + (0.923 - 0.382i)T \)
83 \( 1 + (0.111 + 0.993i)T \)
89 \( 1 + (0.781 + 0.623i)T \)
97 \( 1 + (0.985 + 0.167i)T \)
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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

−22.82241707780314584776975168426, −21.86520312995048484741731729731, −21.17053020157896983177957133697, −20.42646378842615765451899525235, −19.359771511883466692596247803525, −18.90950432686075242079610704375, −18.37577302795725538791911596008, −17.31419793628301958783048149056, −16.200425995930574819418981039113, −15.63392389608529892520278128107, −14.684263045548879253565826898536, −14.53576833333423571177285203878, −12.95750908871926419203946925823, −11.98594155947291094101049776505, −10.975408244743951418837095525133, −10.05833939107209812306098947770, −9.64871790033787151317725950608, −8.65986788336019695308576635407, −7.87700500439057238057562355813, −6.97617611839716982737780571238, −6.18791734867258639870226627076, −4.81232325628935933641571056163, −3.45938736303245881728220114279, −2.40763776781300314105554908872, −2.10059766785163214762843549534, 0.548753419052283207259562431279, 1.32555329771827985374125579791, 2.721012130540083855165424551010, 3.41061830179124654686430224368, 4.79185033001784460881875072924, 6.191089257192639032256609938817, 7.15012142643556707891129300871, 8.01094349546005336024715855740, 8.47203909153497556972925230448, 9.45366249442877026591563004394, 10.141815352128278161250033907052, 11.15820878928833064358765590721, 12.282110268985786843694949533782, 13.10308965531965250194169749066, 13.57333731191000363649953216496, 14.95966592351055920792765115610, 15.75527901839050151316447903870, 16.49045289135662089568463007069, 17.41443171329026580669985688698, 17.97413650063031123817233628908, 19.2638958791987952274241304146, 19.58000134290866518037972036594, 20.227611394622918801140043353403, 20.98970175420193653722975022393, 21.65969231725314095488471673873

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