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Properties

Label 1-731-731.190-r0-0-0
Degree $1$
Conductor $731$
Sign $0.268 - 0.963i$
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 + (0.0186 − 0.999i)3-s + (0.974 + 0.222i)4-s + (−0.892 − 0.450i)5-s + (−0.130 + 0.991i)6-s + (0.608 − 0.793i)7-s + (−0.943 − 0.330i)8-s + (−0.999 − 0.0373i)9-s + (0.836 + 0.547i)10-s + (0.578 + 0.815i)11-s + (0.240 − 0.970i)12-s + (−0.563 + 0.826i)13-s + (−0.693 + 0.720i)14-s + (−0.467 + 0.884i)15-s + (0.900 + 0.433i)16-s + ⋯
L(s)  = 1  + (−0.993 − 0.111i)2-s + (0.0186 − 0.999i)3-s + (0.974 + 0.222i)4-s + (−0.892 − 0.450i)5-s + (−0.130 + 0.991i)6-s + (0.608 − 0.793i)7-s + (−0.943 − 0.330i)8-s + (−0.999 − 0.0373i)9-s + (0.836 + 0.547i)10-s + (0.578 + 0.815i)11-s + (0.240 − 0.970i)12-s + (−0.563 + 0.826i)13-s + (−0.693 + 0.720i)14-s + (−0.467 + 0.884i)15-s + (0.900 + 0.433i)16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.268 - 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.268 - 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(731\)    =    \(17 \cdot 43\)
Sign: $0.268 - 0.963i$
Analytic conductor: \(3.39474\)
Root analytic conductor: \(3.39474\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{731} (190, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 731,\ (0:\ ),\ 0.268 - 0.963i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6698278603 - 0.5087114653i\)
\(L(\frac12)\) \(\approx\) \(0.6698278603 - 0.5087114653i\)
\(L(1)\) \(\approx\) \(0.6337308964 - 0.2891004075i\)
\(L(1)\) \(\approx\) \(0.6337308964 - 0.2891004075i\)

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 \( 1 + (0.0186 - 0.999i)T \)
5 \( 1 + (-0.892 - 0.450i)T \)
7 \( 1 + (0.608 - 0.793i)T \)
11 \( 1 + (0.578 + 0.815i)T \)
13 \( 1 + (-0.563 + 0.826i)T \)
19 \( 1 + (0.999 - 0.0373i)T \)
23 \( 1 + (0.995 - 0.0933i)T \)
29 \( 1 + (0.720 + 0.693i)T \)
31 \( 1 + (0.970 + 0.240i)T \)
37 \( 1 + (0.793 - 0.608i)T \)
41 \( 1 + (0.483 - 0.875i)T \)
47 \( 1 + (-0.974 - 0.222i)T \)
53 \( 1 + (0.185 - 0.982i)T \)
59 \( 1 + (0.330 + 0.943i)T \)
61 \( 1 + (0.240 + 0.970i)T \)
67 \( 1 + (0.733 + 0.680i)T \)
71 \( 1 + (-0.995 - 0.0933i)T \)
73 \( 1 + (0.836 - 0.547i)T \)
79 \( 1 + (0.793 + 0.608i)T \)
83 \( 1 + (-0.916 - 0.399i)T \)
89 \( 1 + (-0.930 + 0.365i)T \)
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

−22.54848674320772363312898051949, −21.775515472041793962871393294489, −20.98617276751423414556674823801, −20.103274283774455138414635752046, −19.45131524558510942790509005596, −18.69291754846674664607988875159, −17.76577675251636755388983957735, −16.98583089482710545748893865666, −16.092358750266397114714132918369, −15.40938163926222461099596338407, −14.92783043278554758485311697460, −14.10984953208762240804348491043, −12.279978716707093004067874175071, −11.39242240141497114333688668492, −11.192446374419292543428989968691, −10.03074096623234637888040107954, −9.30695246176863110882356703040, −8.253917043717948583134748568865, −7.94783212679282755078879614424, −6.56763387813065888440213223982, −5.60410603722618181237221016999, −4.6170697775762125601012434845, −3.18046989269695401736762769249, −2.728440107107521327895953386087, −0.8718061187983690038668168090, 0.851529072082271445991055417372, 1.53305950000322609683518279866, 2.760523197207810135060267770621, 4.03512429665325430439519900734, 5.16637261565196734232319633755, 6.858266713654191832174454177760, 7.099272184932427426065069609149, 7.92936870352135918150168252091, 8.729334576941202166125444726542, 9.601958471847696876265325314183, 10.83454578908090266799441563600, 11.70450012409468879970127117012, 12.053161978509559695123127693948, 13.03896784387164956448733172267, 14.27476043285988798173931578684, 14.94158237798954840852519471605, 16.18738852387145825122566906030, 16.86155990057997382475764969884, 17.57454954731350158147281347944, 18.21706961818326566959995262533, 19.44144460029934346599477333455, 19.54947697203599395990291979465, 20.41187353246252389667816910677, 21.10743556944443153464574530963, 22.606458147684551784732121968491

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