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Properties

Label 1-731-731.22-r0-0-0
Degree $1$
Conductor $731$
Sign $0.407 - 0.913i$
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 + (−0.483 − 0.875i)3-s + (0.974 + 0.222i)4-s + (−0.998 + 0.0560i)5-s + (−0.382 − 0.923i)6-s + (0.923 − 0.382i)7-s + (0.943 + 0.330i)8-s + (−0.532 + 0.846i)9-s + (−0.998 − 0.0560i)10-s + (0.815 − 0.578i)11-s + (−0.276 − 0.960i)12-s + (−0.433 − 0.900i)13-s + (0.960 − 0.276i)14-s + (0.532 + 0.846i)15-s + (0.900 + 0.433i)16-s + ⋯
L(s)  = 1  + (0.993 + 0.111i)2-s + (−0.483 − 0.875i)3-s + (0.974 + 0.222i)4-s + (−0.998 + 0.0560i)5-s + (−0.382 − 0.923i)6-s + (0.923 − 0.382i)7-s + (0.943 + 0.330i)8-s + (−0.532 + 0.846i)9-s + (−0.998 − 0.0560i)10-s + (0.815 − 0.578i)11-s + (−0.276 − 0.960i)12-s + (−0.433 − 0.900i)13-s + (0.960 − 0.276i)14-s + (0.532 + 0.846i)15-s + (0.900 + 0.433i)16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.407 - 0.913i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.407 - 0.913i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(731\)    =    \(17 \cdot 43\)
Sign: $0.407 - 0.913i$
Analytic conductor: \(3.39474\)
Root analytic conductor: \(3.39474\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{731} (22, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 731,\ (0:\ ),\ 0.407 - 0.913i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.863827934 - 1.209719517i\)
\(L(\frac12)\) \(\approx\) \(1.863827934 - 1.209719517i\)
\(L(1)\) \(\approx\) \(1.554624263 - 0.4751598018i\)
\(L(1)\) \(\approx\) \(1.554624263 - 0.4751598018i\)

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 \( 1 + (-0.483 - 0.875i)T \)
5 \( 1 + (-0.998 + 0.0560i)T \)
7 \( 1 + (0.923 - 0.382i)T \)
11 \( 1 + (0.815 - 0.578i)T \)
13 \( 1 + (-0.433 - 0.900i)T \)
19 \( 1 + (0.532 + 0.846i)T \)
23 \( 1 + (-0.815 + 0.578i)T \)
29 \( 1 + (-0.276 - 0.960i)T \)
31 \( 1 + (0.960 - 0.276i)T \)
37 \( 1 + (0.382 - 0.923i)T \)
41 \( 1 + (-0.875 - 0.483i)T \)
47 \( 1 + (-0.974 - 0.222i)T \)
53 \( 1 + (0.943 - 0.330i)T \)
59 \( 1 + (-0.330 - 0.943i)T \)
61 \( 1 + (-0.276 + 0.960i)T \)
67 \( 1 + (0.222 - 0.974i)T \)
71 \( 1 + (0.815 + 0.578i)T \)
73 \( 1 + (-0.998 + 0.0560i)T \)
79 \( 1 + (0.382 + 0.923i)T \)
83 \( 1 + (-0.111 - 0.993i)T \)
89 \( 1 + (0.781 + 0.623i)T \)
97 \( 1 + (0.167 - 0.985i)T \)
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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

−22.52900078336153684471412570098, −21.94288545538891150588280989975, −21.29955374664175956203845159009, −20.221799162590154379928312662210, −19.98546367837540066724023056501, −18.7470198272844233624368484575, −17.58850431948273524938719016473, −16.642255309063908312530021441175, −16.02235093848042377772675714538, −15.03244263456250003688803209969, −14.78733690535857625609740132971, −13.85320236698409903547659260213, −12.3459691237120609534804296177, −11.7961079293161899921566920772, −11.433414590271708534671864120742, −10.45545442988794529208975911809, −9.36637310657340736570395793341, −8.31329056198495671309562611805, −7.12305803105067677254907088923, −6.3618393324655647105770414526, −4.986199565848694874131142064076, −4.644192596133800648152577947918, −3.873367746386783609514574002087, −2.77072726661616238112899160980, −1.37718619959445707949960248746, 0.902978347282334637318909315876, 2.03800992928072681336719899080, 3.33908799081803689714645400935, 4.204746511724220692919915385482, 5.23258035296462444198595916389, 6.02461313552434373149031224035, 7.068890988842787685719446544161, 7.84240278909134654970170478232, 8.23442160982027472483032389730, 10.25067663796814550219179598521, 11.233282496439094707913329128043, 11.74923262256101495039070594227, 12.26929031268452524541509217726, 13.349989536305371995842762099547, 14.10205612918529373087626864756, 14.79864569390356220314463213426, 15.74964694188733508807192468104, 16.68351852983383732624199764430, 17.310629301958071698440631913776, 18.32153800705846715347122987722, 19.41071172213734793199761427798, 19.897435730414246518583151310126, 20.73709890166946208465862676358, 21.83999036104462205250281526512, 22.68920543787311185130631204629

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