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Properties

Label 1-731-731.186-r0-0-0
Degree $1$
Conductor $731$
Sign $0.585 - 0.810i$
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  + (0.623 − 0.781i)2-s + (0.988 − 0.149i)3-s + (−0.222 − 0.974i)4-s + (−0.826 + 0.563i)5-s + (0.5 − 0.866i)6-s + (0.5 + 0.866i)7-s + (−0.900 − 0.433i)8-s + (0.955 − 0.294i)9-s + (−0.0747 + 0.997i)10-s + (0.222 − 0.974i)11-s + (−0.365 − 0.930i)12-s + (0.0747 + 0.997i)13-s + (0.988 + 0.149i)14-s + (−0.733 + 0.680i)15-s + (−0.900 + 0.433i)16-s + ⋯
L(s)  = 1  + (0.623 − 0.781i)2-s + (0.988 − 0.149i)3-s + (−0.222 − 0.974i)4-s + (−0.826 + 0.563i)5-s + (0.5 − 0.866i)6-s + (0.5 + 0.866i)7-s + (−0.900 − 0.433i)8-s + (0.955 − 0.294i)9-s + (−0.0747 + 0.997i)10-s + (0.222 − 0.974i)11-s + (−0.365 − 0.930i)12-s + (0.0747 + 0.997i)13-s + (0.988 + 0.149i)14-s + (−0.733 + 0.680i)15-s + (−0.900 + 0.433i)16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.585 - 0.810i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.585 - 0.810i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(731\)    =    \(17 \cdot 43\)
Sign: $0.585 - 0.810i$
Analytic conductor: \(3.39474\)
Root analytic conductor: \(3.39474\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{731} (186, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 731,\ (0:\ ),\ 0.585 - 0.810i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.355865548 - 1.204814303i\)
\(L(\frac12)\) \(\approx\) \(2.355865548 - 1.204814303i\)
\(L(1)\) \(\approx\) \(1.726714511 - 0.6831007890i\)
\(L(1)\) \(\approx\) \(1.726714511 - 0.6831007890i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 \)
43 \( 1 \)
good2 \( 1 + (0.623 - 0.781i)T \)
3 \( 1 + (0.988 - 0.149i)T \)
5 \( 1 + (-0.826 + 0.563i)T \)
7 \( 1 + (0.5 + 0.866i)T \)
11 \( 1 + (0.222 - 0.974i)T \)
13 \( 1 + (0.0747 + 0.997i)T \)
19 \( 1 + (0.955 + 0.294i)T \)
23 \( 1 + (0.733 + 0.680i)T \)
29 \( 1 + (0.988 + 0.149i)T \)
31 \( 1 + (-0.365 - 0.930i)T \)
37 \( 1 + (0.5 - 0.866i)T \)
41 \( 1 + (-0.623 + 0.781i)T \)
47 \( 1 + (-0.222 - 0.974i)T \)
53 \( 1 + (0.0747 - 0.997i)T \)
59 \( 1 + (-0.900 + 0.433i)T \)
61 \( 1 + (-0.365 + 0.930i)T \)
67 \( 1 + (0.955 + 0.294i)T \)
71 \( 1 + (0.733 - 0.680i)T \)
73 \( 1 + (-0.0747 - 0.997i)T \)
79 \( 1 + (0.5 + 0.866i)T \)
83 \( 1 + (-0.988 + 0.149i)T \)
89 \( 1 + (-0.988 + 0.149i)T \)
97 \( 1 + (0.222 - 0.974i)T \)
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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

−22.92939116307556112862440754408, −21.9039842559495728533274557321, −20.80668385917043221700370487045, −20.299299565646083579198238219107, −19.86728660128055355415254333205, −18.51183139027064699188301735088, −17.55320436172777294371260283832, −16.786606321164000819083989629116, −15.73473158880796325380804736918, −15.363360430605460597437904725388, −14.4614701111513385522670831927, −13.80224603568023153326117589849, −12.83755897679373148792722563285, −12.315132594013936567019391770675, −11.08391303898126794294232903155, −9.91705999267121720349930799431, −8.84537118759857254081097406200, −8.08067329268146606072540810585, −7.48293883840407622760798402255, −6.79805252903039052671956872976, −5.00977669709469512559240703845, −4.61847026630630208554328774047, −3.66757648416053307867328884601, −2.85776546403813587686479850853, −1.186582700038034542907564943042, 1.22895441730423660247323605074, 2.33126102599471727314944059385, 3.180334609813106405701842902683, 3.850922328110851276999819011554, 4.88782910939189650252474453555, 6.09202578063690624547666779311, 7.112414901199981059490539005126, 8.22613612942531173785068920004, 8.98554115734053111524762396422, 9.8052066474943250068927137270, 11.095184919065067048933552423082, 11.60964309400121514192074308086, 12.35479994110851713772059799955, 13.502068356258114679272364531901, 14.14710136731828023706825478847, 14.82259844820564795735093317600, 15.47503383141446178124285484353, 16.34298439935544325579472465954, 18.2513405315099693441942942285, 18.522373099323416320960540562886, 19.39643533429622610814190115213, 19.788916844263130600152421357556, 20.940929796370633410183526163840, 21.49208734279540544526406151686, 22.12740509948160589093943305435

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