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Properties

Label 1-731-731.144-r0-0-0
Degree $1$
Conductor $731$
Sign $0.999 + 0.00107i$
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  + (−0.974 + 0.222i)2-s + (0.0373 + 0.999i)3-s + (0.900 − 0.433i)4-s + (−0.804 − 0.593i)5-s + (−0.258 − 0.965i)6-s + (−0.965 − 0.258i)7-s + (−0.781 + 0.623i)8-s + (−0.997 + 0.0747i)9-s + (0.916 + 0.399i)10-s + (0.943 − 0.330i)11-s + (0.467 + 0.884i)12-s + (−0.365 + 0.930i)13-s + (0.999 + 0.0373i)14-s + (0.563 − 0.826i)15-s + (0.623 − 0.781i)16-s + ⋯
L(s)  = 1  + (−0.974 + 0.222i)2-s + (0.0373 + 0.999i)3-s + (0.900 − 0.433i)4-s + (−0.804 − 0.593i)5-s + (−0.258 − 0.965i)6-s + (−0.965 − 0.258i)7-s + (−0.781 + 0.623i)8-s + (−0.997 + 0.0747i)9-s + (0.916 + 0.399i)10-s + (0.943 − 0.330i)11-s + (0.467 + 0.884i)12-s + (−0.365 + 0.930i)13-s + (0.999 + 0.0373i)14-s + (0.563 − 0.826i)15-s + (0.623 − 0.781i)16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00107i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00107i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(731\)    =    \(17 \cdot 43\)
Sign: $0.999 + 0.00107i$
Analytic conductor: \(3.39474\)
Root analytic conductor: \(3.39474\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{731} (144, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 731,\ (0:\ ),\ 0.999 + 0.00107i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4725646605 + 0.0002535585167i\)
\(L(\frac12)\) \(\approx\) \(0.4725646605 + 0.0002535585167i\)
\(L(1)\) \(\approx\) \(0.4981997076 + 0.1181913645i\)
\(L(1)\) \(\approx\) \(0.4981997076 + 0.1181913645i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 \)
43 \( 1 \)
good2 \( 1 + (-0.974 + 0.222i)T \)
3 \( 1 + (0.0373 + 0.999i)T \)
5 \( 1 + (-0.804 - 0.593i)T \)
7 \( 1 + (-0.965 - 0.258i)T \)
11 \( 1 + (0.943 - 0.330i)T \)
13 \( 1 + (-0.365 + 0.930i)T \)
19 \( 1 + (-0.997 - 0.0747i)T \)
23 \( 1 + (-0.185 + 0.982i)T \)
29 \( 1 + (-0.999 - 0.0373i)T \)
31 \( 1 + (-0.467 - 0.884i)T \)
37 \( 1 + (0.965 - 0.258i)T \)
41 \( 1 + (-0.846 - 0.532i)T \)
47 \( 1 + (0.900 - 0.433i)T \)
53 \( 1 + (0.930 - 0.365i)T \)
59 \( 1 + (0.781 + 0.623i)T \)
61 \( 1 + (0.467 - 0.884i)T \)
67 \( 1 + (0.0747 - 0.997i)T \)
71 \( 1 + (-0.185 - 0.982i)T \)
73 \( 1 + (0.916 - 0.399i)T \)
79 \( 1 + (0.965 + 0.258i)T \)
83 \( 1 + (-0.680 + 0.733i)T \)
89 \( 1 + (0.733 + 0.680i)T \)
97 \( 1 + (-0.330 - 0.943i)T \)
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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

−22.52599666438053509677773814928, −21.89977961632141476194446740049, −20.188920756640295670454038474388, −20.005412903704864251832432629584, −19.09440433258648117237803322260, −18.70707163071888969897895827321, −17.84256998440674919632638164338, −16.98096198366901082103189954559, −16.24533772480875699475725990060, −15.09357969412772587038006590912, −14.61955090016823447470977383354, −13.03403987723399084750137408594, −12.438060982322778176560069301949, −11.79234075697601444379842962810, −10.86012884106484206365379140430, −9.99968761079183677315076938872, −8.88563524748930960042573726604, −8.20381662791440466279082114858, −7.20220192538492393668661040646, −6.72089562375528789698111139811, −5.89098912079090669490502396164, −3.90239670638261596552922121969, −2.96278412089785221866425654974, −2.198756465695749408831701666711, −0.75814766503127946505623109356, 0.45527582836179385432308700444, 2.09080613430038907560806570142, 3.55478066442344520669275686808, 4.106782453084461523922582267723, 5.45747503382834107066666113740, 6.43481877615743356169749032267, 7.37557164197752878329281004785, 8.45094972604987875884773208652, 9.28485946442000588915841639965, 9.56677324406541220153372964372, 10.78987579276967635071338830477, 11.493169599921976130666941029003, 12.22832184220750693664550211395, 13.58827743371184226372359720155, 14.8003725590880306552944997054, 15.33034227682843345793916159043, 16.27498538592847556023608820197, 16.76668955771381592132136805732, 17.1286949540209397827681338015, 18.70480520372475436249910159045, 19.522558063557392419295006485320, 19.79391900283431512679963806733, 20.69305238897310245320593432436, 21.6027574583193339892362221828, 22.464089424025497123932052845126

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