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Properties

Label 1-731-731.419-r0-0-0
Degree $1$
Conductor $731$
Sign $0.828 + 0.560i$
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  + (0.846 − 0.532i)2-s + (−0.578 − 0.815i)3-s + (0.433 − 0.900i)4-s + (−0.960 − 0.276i)5-s + (−0.923 − 0.382i)6-s + (−0.382 + 0.923i)7-s + (−0.111 − 0.993i)8-s + (−0.330 + 0.943i)9-s + (−0.960 + 0.276i)10-s + (−0.998 + 0.0560i)11-s + (−0.985 + 0.167i)12-s + (−0.781 − 0.623i)13-s + (0.167 + 0.985i)14-s + (0.330 + 0.943i)15-s + (−0.623 − 0.781i)16-s + ⋯
L(s)  = 1  + (0.846 − 0.532i)2-s + (−0.578 − 0.815i)3-s + (0.433 − 0.900i)4-s + (−0.960 − 0.276i)5-s + (−0.923 − 0.382i)6-s + (−0.382 + 0.923i)7-s + (−0.111 − 0.993i)8-s + (−0.330 + 0.943i)9-s + (−0.960 + 0.276i)10-s + (−0.998 + 0.0560i)11-s + (−0.985 + 0.167i)12-s + (−0.781 − 0.623i)13-s + (0.167 + 0.985i)14-s + (0.330 + 0.943i)15-s + (−0.623 − 0.781i)16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.828 + 0.560i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.828 + 0.560i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(731\)    =    \(17 \cdot 43\)
Sign: $0.828 + 0.560i$
Analytic conductor: \(3.39474\)
Root analytic conductor: \(3.39474\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{731} (419, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 731,\ (0:\ ),\ 0.828 + 0.560i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5444809647 + 0.1667958572i\)
\(L(\frac12)\) \(\approx\) \(0.5444809647 + 0.1667958572i\)
\(L(1)\) \(\approx\) \(0.7920347226 - 0.3835029442i\)
\(L(1)\) \(\approx\) \(0.7920347226 - 0.3835029442i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 \)
43 \( 1 \)
good2 \( 1 + (0.846 - 0.532i)T \)
3 \( 1 + (-0.578 - 0.815i)T \)
5 \( 1 + (-0.960 - 0.276i)T \)
7 \( 1 + (-0.382 + 0.923i)T \)
11 \( 1 + (-0.998 + 0.0560i)T \)
13 \( 1 + (-0.781 - 0.623i)T \)
19 \( 1 + (0.330 + 0.943i)T \)
23 \( 1 + (0.998 - 0.0560i)T \)
29 \( 1 + (-0.985 + 0.167i)T \)
31 \( 1 + (0.167 + 0.985i)T \)
37 \( 1 + (0.923 - 0.382i)T \)
41 \( 1 + (0.815 + 0.578i)T \)
47 \( 1 + (-0.433 + 0.900i)T \)
53 \( 1 + (-0.111 + 0.993i)T \)
59 \( 1 + (-0.993 - 0.111i)T \)
61 \( 1 + (-0.985 - 0.167i)T \)
67 \( 1 + (0.900 + 0.433i)T \)
71 \( 1 + (-0.998 - 0.0560i)T \)
73 \( 1 + (-0.960 - 0.276i)T \)
79 \( 1 + (0.923 + 0.382i)T \)
83 \( 1 + (-0.532 + 0.846i)T \)
89 \( 1 + (-0.974 + 0.222i)T \)
97 \( 1 + (0.745 + 0.666i)T \)
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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

−22.58356943025262504346755576118, −21.89358869241406521563449305350, −20.96636663380372438114467326356, −20.29874330747705965868016596817, −19.42529207467470685947925594261, −18.205584889199547522252961580974, −17.05496197664718564644934293444, −16.6401410536121815715253933932, −15.7876473003918379707204999062, −15.21296765335064553975230136617, −14.48562883838471568497676207330, −13.3616389844273445835971600959, −12.617197799260812370064639727, −11.462657632148208486582719377198, −11.14605694365195117624196579858, −10.07943053534813128685861355017, −8.96239608597165656638499323782, −7.61708924010346155587865840890, −7.15813815758695314048735508750, −6.15211606924741999486881308675, −4.95296205160907813521238127065, −4.446899253050993595501560187587, −3.53560386477099990606985340236, −2.71858793843872994888227887014, −0.23005937450071192006322080259, 1.20986875276805926504440883384, 2.53330062329451971989450948072, 3.18984814008984062929769260892, 4.68416167890644982541126051462, 5.34520034337296989509089390962, 6.1086951602693556978535239819, 7.306897819921909299667535201936, 7.937253732908783837733528704942, 9.2665624042492881555215391625, 10.47633292370682239170001091916, 11.23127558733560990093433419855, 12.06355612097184774100502245806, 12.72213348402931098956047876177, 12.966102667559590098131264527145, 14.35075045034738463050153049165, 15.19546929374546855172996522201, 15.92689630361506030663692050285, 16.702243227888582689005473581912, 18.106659619855223980657893121001, 18.71438178030677773131601289835, 19.37929212735461697506192605127, 20.089929436450464567468934169567, 21.051663664373284222609649367275, 21.9826597303465400726690384326, 22.82213802931676293451826518082

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