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Properties

Label 1-731-731.437-r0-0-0
Degree $1$
Conductor $731$
Sign $-0.980 - 0.194i$
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  + (0.707 − 0.707i)2-s + (−0.608 − 0.793i)3-s − i·4-s + (0.793 − 0.608i)5-s + (−0.991 − 0.130i)6-s + (0.793 + 0.608i)7-s + (−0.707 − 0.707i)8-s + (−0.258 + 0.965i)9-s + (0.130 − 0.991i)10-s + (−0.382 − 0.923i)11-s + (−0.793 + 0.608i)12-s + (0.866 − 0.5i)13-s + (0.991 − 0.130i)14-s + (−0.965 − 0.258i)15-s − 16-s + ⋯
L(s)  = 1  + (0.707 − 0.707i)2-s + (−0.608 − 0.793i)3-s − i·4-s + (0.793 − 0.608i)5-s + (−0.991 − 0.130i)6-s + (0.793 + 0.608i)7-s + (−0.707 − 0.707i)8-s + (−0.258 + 0.965i)9-s + (0.130 − 0.991i)10-s + (−0.382 − 0.923i)11-s + (−0.793 + 0.608i)12-s + (0.866 − 0.5i)13-s + (0.991 − 0.130i)14-s + (−0.965 − 0.258i)15-s − 16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.980 - 0.194i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.980 - 0.194i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(731\)    =    \(17 \cdot 43\)
Sign: $-0.980 - 0.194i$
Analytic conductor: \(3.39474\)
Root analytic conductor: \(3.39474\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{731} (437, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 731,\ (0:\ ),\ -0.980 - 0.194i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1912253298 - 1.946754363i\)
\(L(\frac12)\) \(\approx\) \(0.1912253298 - 1.946754363i\)
\(L(1)\) \(\approx\) \(0.9313506106 - 1.130589209i\)
\(L(1)\) \(\approx\) \(0.9313506106 - 1.130589209i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 \)
43 \( 1 \)
good2 \( 1 + (0.707 - 0.707i)T \)
3 \( 1 + (-0.608 - 0.793i)T \)
5 \( 1 + (0.793 - 0.608i)T \)
7 \( 1 + (0.793 + 0.608i)T \)
11 \( 1 + (-0.382 - 0.923i)T \)
13 \( 1 + (0.866 - 0.5i)T \)
19 \( 1 + (0.258 + 0.965i)T \)
23 \( 1 + (-0.991 - 0.130i)T \)
29 \( 1 + (-0.130 - 0.991i)T \)
31 \( 1 + (-0.608 - 0.793i)T \)
37 \( 1 + (-0.608 - 0.793i)T \)
41 \( 1 + (0.923 - 0.382i)T \)
47 \( 1 + iT \)
53 \( 1 + (-0.258 - 0.965i)T \)
59 \( 1 + (0.707 + 0.707i)T \)
61 \( 1 + (-0.793 - 0.608i)T \)
67 \( 1 + (0.5 - 0.866i)T \)
71 \( 1 + (0.991 - 0.130i)T \)
73 \( 1 + (0.130 + 0.991i)T \)
79 \( 1 + (-0.608 + 0.793i)T \)
83 \( 1 + (0.258 + 0.965i)T \)
89 \( 1 + (-0.866 - 0.5i)T \)
97 \( 1 + (-0.923 - 0.382i)T \)
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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

−23.001945977441075450169324672511, −22.001397970335489935768027906481, −21.59997909396364848684883807270, −20.75006555260006349085495494242, −20.192817101341824671491058398213, −18.17067169525881036908199949708, −17.90476343846173005281107255625, −17.15623225657457639570701476934, −16.296054055397184312910808791308, −15.50440310477322570280194104067, −14.70146948001284494107160930686, −14.05929224813172681369742293284, −13.26673225924755027008552037849, −12.142949416828704721985974884645, −11.19801516039240255020920240345, −10.58356467749098766139437146800, −9.54056214241399785078291785305, −8.57511017450522990703666170324, −7.262393119473248960513759658379, −6.67022354039953090353713451834, −5.644153213047483122389279170116, −4.920111792142809019604621822434, −4.122326832425947318359590474858, −3.113203760755709704010717348041, −1.7412845973011258156285375400, 0.79879122166540678916550175985, 1.764473336303521857489453270849, 2.47584310789067636709430185257, 3.92926980973885256411798626064, 5.17445701242144743205929004769, 5.8036340195107423916438313605, 6.13037007530911020225011721069, 7.85307379313622446316684390112, 8.60586903075311245737456268914, 9.794480141799388000459326382321, 10.83367192063835396547130762254, 11.391731384136267857620192664587, 12.36924878172672650572551527108, 12.88501227458600314523612731700, 13.84313862761254405602794160582, 14.20436088226974350873419215663, 15.6234591461002385426289705359, 16.40744614095836311964246889634, 17.543178029252770894008789558097, 18.291040075972184196014418935979, 18.71680590114753989198836645126, 19.81265875922532793004687643838, 20.89667996744631991239086176294, 21.16645252540402165233612534675, 22.18446678098518049999853747233

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